Fortunately, after a frantic last-minute search, a family member of the father located the converted woolshed for the couple to rent until they could find suitable long-term accommodation.
Central Health has been working with the couple.
Youth Services manager Louise Howe said despite being employed, the father's youth and relatively short work history meant he had been unsuccessful in applying for rental properties.
"He's absolutely no risk to a landlord. He's really switched on and he's got a young baby that he wants to support. It's dire."
But they were not the only ones, she said.
"We've had quite a few teen parents contact us [for help] and we do have problems finding homes for them, because they are young and they haven't actually got any referees because they have not rented before.
"We've managed to find some homes for some of them and some of them are living with aunties and that kind of thing."
As far as she knew, Housing NZ had no available properties in CHB.
"We do ring them up. There's just nothing out there."
Howe said even finding a boarding situation for clients was difficult, and the level of social housing in CHB was an ongoing problem.
"Basically, it's non-existent."
CHB district councillor Kelly Annand was familiar with the teenage couple's case as director of the Connecting for Youth Employment Trust.
She described it as a sad situation, and indicative of the problems facing not just CHB, but also much of the country, due to dwindling social housing stock and a lack of available rental properties.
"I really feel for this young couple that they are in this situation.
"There is certainly a conversation that needs to be had within the community and with our central-government partners about what we can do locally to fix this problem."
She suggested a local register could be created to take stock of what emergency-type accommodation was available in the district for vulnerable and at-risk people.
"Some people might have granny flats that they could rent out, or shearers' quarters and things like that. So that might be a good place to start," she said.
A Housing NZ spokesman said last week that all 34 of its properties in Central Hawke's Bay were fully tenanted, and the department had no plans to acquire additional properties.
The Ministry for Social Development's deputy chief executive for housing, Scott Gallacher, said 4865 applications had been made to the national Social Housing register as at the end of the March 2017 quarter, with nine applications coming from CHB.
Nationally, the average waiting time for people to access social housing was 52 days at the end of March, but the department was not able to provide a breakdown of waiting times in CHB before our print deadline.
The department granted 9218 Emergency Housing Special Needs Grants across the country up to the March 2017 quarter, with six granted in CHB.
People in immediate urgent need of somewhere to stay, or in the next seven days, can call MSD on 0800 559 009.