Deborah Wai Kapohe said it meant her family would be able to move to a warmer, drier home if mediation failed to resolve their claim before winter.
The "fantastic" offer came after a story on the family's plight was published in yesterday's Otago Daily Times. They were also offered a caravan for the winter.
"We're really a lot happier," Ms Wai Kapohe said.
The offer of the home was made by the family's Dunedin doctor, who remained anonymous.
Ms Wai Kapohe and husband Michael Beazley bought their $550,000 home in Glenleith late in 2010, after checking it had a code of compliance certificate issued by the Dunedin City Council.
However, they soon discovered the first signs of cracks, leaks and toxic mould, followed by a list of other faults, and last April their house was declared a leaky home.
The couple and their two children spent last winter living in one room of their home, trying to keep warm while waiting for the outcome of mediation aimed at reaching a settlement with the council and other parties.
The hearing had been expected before Christmas last year, but a date was now set down for late March, and an adjudication date would be needed later in the year if mediation failed, Ms Wai Kapohe said.
Yesterday, she said the couple could now move out of their leaky home, but continue making mortgage payments, if mediation talks failed and adjudication was needed.
Their doctor had been following their case in the ODT and was considering making the offer before yesterday's story , she said.
"He said that for a while he'd been thinking about it, but [yesterday's] story really prompted him to give [us] a call.
"He, more than anyone, knows what Michael has been going through."
Mr Beazley had suffered health problems believed to be related to the stress of the situation over winter, with glandular fever, blood poisoning, the flu and asthma "all at the same time", Ms Wai Kapohe said.
"It's the burden that it lifts off us, of going through that cold winter, [that makes a] huge difference."
