The child, who came to be known as Baby Peter, was found on the doorstep of the vicarage of St Peter’s Church, in Hough Lane, Bramley, on November 23, 2016.
He was taken to hospital, where he was pronounced dead, but a burial or cremation could not take place while the hunt for his mother was under way.
Detectives now say they have exhausted all lines of inquiry in their efforts to trace the infant's mother and will speak to the Coroner's Office about finally releasing his body for a funeral.
But they have today made a fresh appeal urging the mother to come forward to say a final goodbye to her son and help to lead the arrangements for him to be laid to rest.
Mother's welfare 'remains main concern'
Detective Superintendent Nicola Bryar, who has led the investigation, said: “Sadly we have now explored and exhausted all the available lines of enquiry without being able to identify the mother of this baby boy.
“Throughout this investigation, her welfare has remained our main concern. The immediate concerns we had for her health have lessened with the passage of time but we are still keen to trace her and make sure she has the help and support she needs after what has clearly been a traumatic experience.
“On the first anniversary of her son being found I released a special video message to appeal directly to her to contact me.
“The sentiments I expressed in that message a year ago remain exactly the same today. She is not in any trouble with the police and there is still an opportunity for her to come forward.
“Our discussions with the Coroner’s Office will include when her son’s body can be released for a funeral. That is something we would like her to be able to lead the arrangements for and to have the chance to say goodbye properly.
“It is not too late for her to make herself known to me and I hope she might feel able to do that now.”
Baby Peter, who was white, was found inside a Tesco Bag for Life and had been wrapped in a distinctive blue striped towel and a men’s white 15-inch collar shirt from clothing firm Dunnes.
It was quickly established that the infant had reached full term and the placenta was still attached, so in all likelihood he had been left at the church soon after he was born.
It remains a mystery whether it was the mother herself or someone else who left the baby on the steps of the vicarage, which is not easily identifiable to those unfamiliar with the area.
The infant appeared lifeless and was cold to the touch but was given CPR at the scene by paramedics before being taken to hospital, where he was pronounced dead.
A post mortem examination and further work by the pathologist was not able to ascertain the cause of death or whether he was already dead when he was left.
His death is not being treated as suspicious and his body is still being retained while officers liaise with the Coroner’s Office.
The tragic events prompted an outpouring of tributes from the local community, with flowers, teddy bears and notes left outside the church as news of the baby’s death became more widely known.
The Rev Paul Crabb, who found Baby Peter, led prayers for the child and his mother and, a year on, reached out to the mother in an emotional message, saying the people of Bramley "have not forgotten your baby or you".
Police find mother's DNA
West Yorkshire Police set out the efforts it had made to trace the boy's mother before concluding that it had exhausted all lines of inquiry.
The force said officers had carried out extensive enquiries over the last two years including contacting schools, health visitors and midwife services.
Media appeals were made - including a video appeal on the first anniversary of Baby Peter's death - and a mass leafleting and poster campaign was conducted in the area.
Detectives also trawled CCTV around the vicarage and wider area and viewed several hours of footage.
A DNA profile for the mother was recovered but is not recorded on the police database. The DNA found on the shirt suggested it had been worn by a female, the force said.
A dedicated mobile phone number for the investigation has now changed and is 07881 894505.
Any information can also be passed to the police via 101, quoting Operation Plaindance, or by visiting www.westyorkshire.police.uk/contact-us.