Constance Marten and her former convict partner Mark Gordon are feared to be camping in sub-zero temperatures with their new-born baby.
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A new CCTV image shows the couple with their baby in Newhaven, East Sussex.
Addressing Constance and Mark directly, Met Police Detective Superintendent Lewis Basford said: “Constance and Mark, your baby has spent the first month of its short life exposed to the elements when it should be safe and warm and, most importantly, seen by medics.
“After a month you must be running low on cash. Please pick up the phone and let us know, at the very least, that you are okay.
“We are ready to come to you and see that you and the baby get medical attention.”
He stressed the force “know the baby was still alive” on January 8.
Cops fear the pair have been camping in nearby countryside – including at times when the temperature plummeted to sub-zero.
The couple were last seen with their baby, in Newhaven, East Sussex on January 9.
The family have been missing since their car broke down and burst into flames on the M61 on January 5.
Shortly before going missing, Constance, 35, gave birth to Gordon’s child “with no medical attention”.
They they then travelled to Liverpool and then onwards to Harwich, Essex, via taxi at around 3.30am on January 6.
The couple were seen by a member of the public in Harwich on January 7, police say.
Officers also received a number of confirmed sightings of the family in Colchester January 6 and 7.
The couple were then placed near East Ham Station in London on January 7.
They are believed to have been sleeping rough after being spotted buying camping equipment from an East London Argos on January 7.
The same day, they were seen on grainy CCTV dumping the baby’s pram in the port town at 11.46pm.
Cops fear they have been camping in sub-zero temperatures.
New CCTV shows the pair walking along Cantercrow Hill in Newhaven and into the fields beyond.
They were carrying a number of bags and their blue tent.
Cops fear they were camping in the area, but could have moved elsewhere as time passed.
He added: “For a month now Mark, Constance and her new-born have been continually on the move in sub-zero temperatures with no medical attention and we are increasingly concerned for their wellbeing.
“We know that the baby was still alive on January 8 and finding the baby remains our top priority.
“Maybe you have information but were reluctant to come forward to help us find the family.
“It does not matter why you did not speak to us earlier, what matters is doing the right thing now for the good of this highly vulnerable infant.
“If you have seen the family, if you gave them directions, if they got into your taxi or came into your shop to buy food, drink or nappies – no matter how small a detail it may seem – please do get in touch.
“Please also be assured that if they have paid you to provide a service, a lift, somewhere to stay or something else, you are not in any sort of trouble, we just need you to tell us what you know so that we can find them and make them safe.”
It comes after an old pal revealed that Constance “could have gone very far” as an actress before she became estranged from her aristocratic family.
In fact, Constance was deemed an ‘It girl’ when she appeared on upmarket mag Tatler’s babe of the month page while she was a student in 2008.
The daughter of Napier Martin, a former page to the Queen, and granddaughter of Mary Anna Marten, whose godmother was the Queen Mother, Constance comes from a family of wealthy landowners.
It is believed that she paid for her time at drama school via a multi-million pound trust fund stored at C. Hoare & Co, the UK’s oldest private bank.
However, she has reportedly led an isolated life since moving in with Gordon in 2016.
The 48-year-old is a convicted sex offender who served 20 years in a US prison for rape and battery.
He was convicted for attacking a woman in Florida when he was just 14.
His victim recently revealed that he held her captive and threatened to kill her children if she screamed during the horrific attack.
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