user: He was discovered at about 04:50 BST at Emerson Court in Horden, near Peterlee, and taken to the University Hospital of North Tees in Stockton.
The baby, who was less than 12 hours old, was named Jack by hospital staff. He is "fit and well", Durham Police said.
At a press conference, officers confirmed his mother had come forward.
She has been medically examined and will be interviewed.
Police are working with the county council's children's services unit.
The baby was found by a pet dog who was being taken for a walk.
Mary Cartwright, mayor of Peterlee, saw what happened as she was returning from Newcastle Airport.
She said: "I saw the policeman with a bundle in his arms and I could see the baby's head and it was in a blanket.
"He ran across the road and round to the side of the ambulance and the crew took the baby.
"There was nobody around and then there was this baby and I knew something wasn't quite right," she said
Initial reports said the child had been found on nearby Fourth Street.

Sum: A newborn baby has been found abandoned on the doorstep of a property in County Durham.


Searches were carried out in Belfast, Tyrone and Fermanagh on Thursday.
The man was taken to Dungannon police station for questioning.
Det Ch Insp Ian Wilson said an investigation was launched "following a number of complaints from farmers about the management of their business affairs".
Mr Wilson said police are liaising with the Chartered Accountants Regulatory Board and HM Revenue and Customs.
"The investigation is at an early stage. A substantial amount of documents will need to be examined and our enquiries will continue for some time," he added.

Sum: A 42-year-old man arrested as part of an investigation into fraud and money laundering has been released on police bail pending further enquiries.


The Space Centre in Preston thought it had secured Â£800,000 in grants, only to find they never existed.
The centre said it is in the process of expanding its premises to handle the demand for care in the region.
Lancashire police confirmed it was investigating allegations involving an ex-employee of the centre.
Alison Shorrock, centre manager, said the staff and people who used the facilities had been left "devastated" by the news.
"I was shocked as we thought the remaining grants were in place, but as we rang them one by one, to our absolute horror, we found out that there was absolutely no funding still to come. They had never heard of us.
"We were running around like children at Christmas because we were so excited at the possibilities that the new centre would bring for all the disabled people, and suddenly to have your hopes and dreams crash to the floor was one of the worst moments of my life."
The centre, on Pedders Lane, now say they need to raise Â£500,000 to pay off their immediate debtors, including those involved in building an extension to the building.
About 200 groups a month use the centre's facilities, which include multi-sensory areas.
It opened in 1993, with the company securing the building next door in 2011 to expand and build a further sensory room, due to open shortly.
A Lancashire Police spokesperson said: "We have been made aware of a report regarding financial irregularities made in relation to an ex-employee and inquiries are ongoing to establish the full facts."

Sum: A Lancashire centre which provides services for disabled people faces possible closure after allegations of "financial irregularities".


Spy Booth depicted 1950s-style spies "snooping" on a man in a phone box. It appeared in 2014, three miles from government listening post GCHQ.
A pile of rubble now lies on the ground below the wall of the house it was on.
The piece gained listed status last year after Cheltenham Borough Council granted it retrospective planning permission.
A social media post on Saturday appeared to show the mural being cut down behind a tarpaulin, with the sound of machinery audible.
Local people had campaigned to save the artwork, on the corner of Fairview Road and Hewlett Road, after an apparent attempt to remove it for it to be sold in July 2014.
It was later defaced with spray paint and has been vandalised several times since.
In January this year the Grade II-listed house was put for for sale.
A Gloucestershire Police spokeswoman said the force had had no reports of a crime being committed.
Looking behind the tarpaulin and scaffolding which had already been erected around the Banksy, all that can be seen now is a bare brick wall and a pile of rubble.
But has the Banksy artwork actually been destroyed?
Bits of the rubble definitely have plaster on them, but there do not appear to be many pieces with paint on them.
So could Spy Booth have in fact been taken away and saved?
A post on Twitter from a local resident suggests it is not the last we have seen of the mural.
"It'll be back. Plans have been made," the post says.
Or perhaps that simply means it will be repainted one day?
All attempts I have made today to contact the house owner have failed.
Until we find out exactly what has happened to Spy Booth, it will remain a mystery.
Steve Jordan, leader of Cheltenham Borough Council, said work had been taking place to repair plasterwork on the wall of the house after the council had issued an enforcement notice.
But he said he was unaware the Banksy had vanished.
"It is protected by a listing. I will have a look at what the situation is, certainly," he said.

Sum:

assistant: A Banksy artwork painted on the side of a listed building in Cheltenham has been removed.