Malone, New York (CNN)[Breaking news, posted at 4:03 p.m. ET]

Escaped murderer David Sweat has been shot and is in custody, law enforcement sources told CNN on Sunday. Sweat was transported to the Alice Hyde Medical Center in Malone, New York, after being shot, an officer at the hospital said. Sweat appears to be the only person shot during his apprehension, the officer said.

[Original story, posted at 2:59 p.m. ET]

If not for some gunfire and his urge to cough, escaped murderer Richard Matt -- who was shot dead Friday -- might not have found himself on the end of three bullets to the head.

With prison-break partner David Sweat, also a convicted murderer, still on the lam Saturday, New York state officials are counting on another misstep.

\"He's been on the run now for three weeks,\" Franklin County Sheriff Kevin Mulverhill said of Sweat. \"He's tired. He's fatigued. He's hungry. He's going to make a mistake.\"

Two days after a member of a tactical unit opened fire on Matt -- hitting him three times in the head, according to state police -- about 1,200 federal, state and local law enforcement officers were searching vehicles at roadblocks and scouring dense woods in upstate New York for the now lone escaped inmate.

Since Sweat and Matt broke out of the maximum security prison in Dannemora together -- and appeared to stick together on their flight from the law -- authorities were hopeful they could be right on Sweat's heels after encountering Matt.

Searchers had at times followed two sets of footprints, but when they gunned Matt down one day after his 49th birthday, there was no sign of Sweat nearby.

So, on all-terrain vehicles and in helicopters, they continued looking for the man who has eluded them for three weeks, using infrared vision devices to peer through the night.

As of Saturday, however, there was still no sight of Sweat, according to a law enforcement source with knowledge of the investigation. And bloodhounds had not picked up his scent.

There have been \"no sightings ... no real credible leads,\" Mulverhill told CNN. \"The state police do believe they have some evidence that indicates he is within that hard perimeter.\"

That search is focused on a 22-square-mile perimeter along State Route 30 between County Route 41 in the town of Malone and County Route 26 in the town of Duane, the state police said in a statement.

Grid searches were being conducted until the perimeter is cleared, officials said.

\"I don't think the availability of cabins within this perimeter is the same as what they've had until they've gotten to this point,\" Mulverhill said.

Authorities believe the men had been traveling one to two miles a day, Mulverhill said.

A bang

If not for the blast of gunfire, Matt, too, may have continued to elude officers from multiple law enforcement agencies, who, for three weeks, have combed brush, field and forest in the dark green shadows of the rolling Adirondack Mountains.

He may have again stayed out of eyeshot of searchers who locked down schools and homes in surrounding quiet rural towns, while they pursued over 2,000 tips on the fugitives' possible whereabouts.

But then on Friday, owners of a recreational vehicle driving down Route 30 heard a bang that eventually led officers to Matt. At the time, they were near Malone just south of the Canadian border.

\"They thought maybe they had a flat. They got out; they realized they didn't have a flat,\" said New York State Police Superintendent Joseph A. D'Amico.

Matt had taken a shot at the camper heading down Route 30 near Malone, according to Clinton County Sheriff David Favro. The driver saw a bullet hole in the camper. The reason for his firing at the camper is still unclear.

A cabin

So, they drove on, until they reached their campsite, where they got out to take a closer look.

\"They examined the trailer that they were towing and realized that there was a bullet hole through the back of the camper,\" D'Amico said.

They called police, who received additional reports of subsequent gunfire.

The officers who responded to the camper drivers figured that the gunshot they heard probably happened about eight miles back up the road. When they searched there, they came upon a cabin in the town of Duane.

Who are Richard Matt and David Sweat?

A cough

As they stepped inside, the smell of gun smoke hung in air. And it looked to them like someone had just been in the cabin, but fled out the back door, D'Amico said. They pursued.

Outside, something moved. And they heard something that sounded like coughing. It wasn't an animal noise; it was human, D'Amico said. It didn't take long for a customs and border protection tactical team to come face-to-face with Matt.

\"They told him to put up his hands,\" D'Amico said.

Authorities have warned from the start that the pair of fugitives might get their hands on guns. Matt had. And it may have been his undoing at that moment. Matt didn't shoot, but he also didn't comply with officers' order, so they opened fire.

A member of the tactical unit opened fire, hitting Matt three times in the head with a semi-automatic weapon, according to a New York State Police news release Sunday. An autopsy showed he died of \"severe skull fractures and brain injuries.\"

\"We recovered a 20-gauge shotgun from Matt's body at the location,\" said D'Amico.

Matt's body was transported to Albany Medical Center for an autopsy, according to the state police. Toxicology results are pending.

Matt also had \"bug bites on the lower extremities, blisters, and minor abrasions consistent with living in the woods for three weeks,\" according to the state police release. Earlier, a source familiar with the investigation said police on Friday also found Matt's underwear in an old outhouse and a test showed it had his DNA.

Brother relieved he's dead

\"You never want to see anyone lose their life,\" New York Gov. Andrew Cuomo said. \"But I would remind people that Mr. Matt was an escaped murderer from a state prison. Mr. Matt killed two people who we know about. Mr. Matt killed his boss in a dispute and dismembered him. He fled to Mexico and then he killed another person in Mexico and was imprisoned in Mexico.\"

Matt's half-brother was relieved to hear he was dead.

\"Right now I still can't think of him as the Rick that I knew. I can only think of him as the man who threatened to kill me and has killed other people and has escaped,\" Wayne Schimpf told CNN affiliate WKBW.

Great escape

With Matt gone, searchers turned their attention to finding Sweat.

Authorities don't believe Sweat has slipped the perimeter, but Mulverhill said: \"We still don't have any confirmation that he was, in fact, with Matt.\"

They set up the perimeter and led dogs in after him. D'Amico warned residents to be on the lookout. \"Sweat is still out there. He is considered dangerous,\" he said.

He advised anyone spotting the convict not to approach him but to dial 911 instead.

Mulverhill said, \"Mr. Sweat, if he has the opportunity, we believe he will arm himself. I'm sure there are a number of camps out there that they spent the night in or they've broken into that haven't even been reported yet. Could they have come across another weapon? Absolutely. We're certainly not discounting that.\"

Sweat was imprisoned at the Clinton Correctional Facility for shooting dead an officer who pursued him after a robbery he committed.

Behind bars, Sweat and Matt cozied up to a prison tailor and a guard who allegedly brought them tools, which they used to cut their way through a labyrinth of multiple barriers and underground passageways. They emerged to freedom through a manhole in the middle of a street.

Guards discovered them missing on June 6, during a routine bed check at 5:30 a.m.