Chancellor Gordon Brown says he is upbeat about his son's future after revealing four-month-old Fraser has been diagnosed with cystic fibrosis.
Gordon and Sarah Brown pictured after baby James Fraser's birth
Mr Brown received messages of support from political allies and opponents, with Tory leader David Cameron among the first to offer his best wishes.
In a positive statement Mr Brown and his wife Sarah said their son was "fit and healthy".
They lost their daughter Jennifer Jane in 2002 after she was born prematurely.
A spokesman for the couple - who have one other son, John - said they were told in late July that Fraser may have the genetic disease and tests had now confirmed it.
The spokesman said Fraser was "fit, healthy and making all the progress that you would expect any little boy to make".
Cystic fibrosis (CF) is a life-threatening inherited disease that disrupts the way the digestive and respiratory systems work.
Most common inherited life-threatening disease in UK
It affects vital organs - lungs and pancreas
There is currently no cure
More than 7,500 people in UK have CF
70% of those are under 20 years old
Each week five babies are born with CF
Average life expectancy is 31
Figures from the Cystic Fibrosis Trust
Mr Brown's spokesman said: "Thousands of other parents are in the same position.
"They are confident that the advice and treatments available, including proper exercise and, later, sporting activity will keep him fit and healthy.
"The NHS is doing a great job, and Gordon and Sarah are very optimistic that the advances being made in medicine will help him and many others, and they hope to be able to play their part in doing what they can to help others."
Messages of support were issued soon after Fraser's condition was made public.
Conservative leader David Cameron, whose four-year-old son Ivan has cerebral palsy, said: "Sam and I are thinking of Gordon and Sarah and their family at this time and we send them our best wishes for the future."
Housing minister Yvette Cooper, a close friend of the Browns along with her husband, Economic Secretary Ed Balls, told the BBC Fraser was a "happy and healthy little boy".
She added: "They are a very strong and happy family, so whilst obviously, it is the same for any parent, it's not the kind of thing you ever want to happen - but they are very optimistic."
Ed Owen, a friend of Mr Brown's and former political adviser whose four-year-old daughter has cystic fibrosis, explained the treatment Fraser would receive.
"It's a pretty rigorous, heavy duty regime of medication, physiotherapy, regular hospital trips, but within that and alongside that, kids can enjoy a relatively happy lifestyle, and there's every prospect that Gordon and Sarah's son will have that for years to come," he told BBC's Newsnight programme.
Mr Brown's brother, John, said: "The key thing was that the baby arrived safely, given what they have been through."
Mr Brown, who is widely tipped as the favourite to succeed Tony Blair when he steps down as prime minister, said after Fraser's birth in July: "I love being a dad. It's great fun and there's nothing more important and there's nothing I enjoy better. "
Cystic Fibrosis Trust chief executive Rosie Barnes said while the disease remained a "very serious medical condition", advances meant the future was much more optimistic than it used to be.
"I believe Fraser was tested at birth for cystic fibrosis so it would be diagnosed just a few weeks after he was born.
"If that test takes place it's very quick and treatment can start immediately... A child diagnosed at birth and treated immediately should remain quite well."
Ms Barnes said he would have to accept regular medical treatment. She also said that while there were people in their 20s and 30s who were waiting for lung transplants, the treatments today were not available when they were children.
A Downing Street spokesman said anything Mr Blair had said to Mr Brown was a private matter.