Irmantas Kuros’ 12-year-old daughter was also with him in the car when police stopped it in Portadown last week - just days after he had been banned from driving.
Adjourning the 39-year-old Lithuanian national’s application for bail, a judge ordered a full report from social services on his standard of parental care.
Kuros, a former mushroom picker from Clendinning Way in the town, faces charges of driving with excess alcohol, dangerous driving, driving while disqualified and having no insurance.
Prosecutor Kate McKay said police spotted him driving along the Mahon Road with a baby on his lap on October 24.
They pursued his Audi vehicle and saw the child being passed to a passenger, the court heard.
A preliminary breath test carried out after the car was stopped indicated he was three times over the limit, according to Mrs McKay.
The barrister revealed that he had been banned from driving three days previously.
During interviews Kuros said he had gone to pick up his daughter at a bus stop after school.
He took his five-month-old son in the car, claiming to have put him on his lap because the baby enjoyed it.
Mrs McKay stressed: “Not only was he putting his young baby at risk, but also the life of his 12-year-old daughter.”
Defence counsel Damien Halleron said his client could not explain why he was so drunk when stopped.
Despite being newly disqualified Kuros had gone to collect his daughter because his wife is still nervous about driving on the left hand side of the road, the court heard.
Mr Halleron added: “His son became agitated in the back of the car... and then in an effort to calm the child down he took the child on his knee.
“That is an extremely dangerous thing to do and he concedes that in hindsight.”
Following submissions Mr Justice Maguire requested more details on the family’s domestic circumstances before deciding whether to release Kuros from custody.
He said: “This man acted in a way here which was reckless to the welfare of a five-month-old child.”