Gordon Brown has dismissed the attempted challenge to his leadership as a "storm in a teacup".
The prime minister told BBC Radio Solent he was "leading from the front" and would say "what I think" even if at times it proved to be unpopular.
Earlier, his spokesman said Mr Brown had the full support of the cabinet.
On Wednesday, ex-ministers Patricia Hewitt and Geoff Hoon wrote to all Labour MPs asking them to support a call for a secret ballot on Mr Brown.
But few Labour MPs spoke out publicly in support of their move and no cabinet ministers said they backed the call.
Justice Secretary Jack Straw said on Thursday morning the attempted coup had "sunk" and predicted it would unite the party.
But Conservative leader David Cameron said the plot showed Mr Brown's government was "deeply divided" while Lib Dem leader Nick Clegg said Labour had "lost its way" and probably would not resolve its differences "until they are in opposition".
Mr Brown's first public comment on the apparent attempt to unseat him as Labour leader came in a long-scheduled BBC Radio Solent phone-in.
He said in his job "every day there is a new problem, every day there is a new challenge - it is how you deal with it" that was important.
Dealing with weather
He added that if he were to "lose track of... the most important things", such as dealing with the weather problems and Afghanistan, he "would not be doing my job properly".
Asked about events on Wednesday, he said: "I was actually holding meetings, first of all about Afghanistan and about some of the terrorist implications of what had happened in America.
"And then, to be honest, for the rest of the day, I was dealing - as I am this morning - with how we can co-ordinate better our supply for dealing with these emergencies, so it's taken up very little of my time," he said.
Gordon Brown jokes about the plot
"I think it's one of these sidelines in this time when people are far more worried about - as they should be - about what we're doing to deal with the weather and how we're making sure that people are safe and secure.
"So it's not going to take up much of my time and hasn't certainly taken up much of my time today or will in future days."
Asked whether he was concerned about the length of time it took (over six hours) for some cabinet ministers to publicly oppose the ballot Gordon Brown said all had issued statements backing him "within an hour or two".
And at a business event on Thursday afternoon, which had had to be relocated because of the weather, Mr Brown joked that "hand on heart, yesterday I didn't think I would be here today" before adding after a pause "we were due to launch this in Southampton".
Earlier Justice Secretary Jack Straw, who ran Mr Brown's Labour leadership bid when he succeeded Tony Blair in 2007, said: "It is certainly the case that this extraordinary announcement yesterday by Geoff Hoon and Patricia Hewitt has sunk and it's completely ill-judged and very ill-advised and all of us made that clear yesterday."
He said he did not believe the party was still divided over Mr Brown, adding: "I actually think it will have the effect of bringing people behind his leadership even more."
'Getting stuck in'
Mr Straw admitted the story was "something we could easily have done without" and said the party needed to concentrate on making the case for Labour at the next general election - due by June.
The BBC understands that those involved in organising the call for a ballot on Wednesday had believed that up to six cabinet ministers might back them in the "right circumstances".
We've got an election to fight, Gordon is leading us into it, we're determined to win it under his leadership and I'm looking forward to getting stuck into it
BBC political editor Nick Robinson said sources had named Mr Straw as one of the six - although he stressed there was no suggestion or evidence that any of those named was involved in planning the coup attempt and all had issued statements criticising it.
Mr Straw said he was angry that the claim had not been put to him before it was broadcast.
Foreign Secretary David Miliband - whose statement of support came more than six hours after the coup attempt was launched, and was widely seen as lukewarm - said it was "not right" to suggest that he had supported the bid.
He added: "I spent yesterday like other members of the government on the business of government. Today, we're getting on with the business of government.
"We've got an election to fight, Gordon is leading us into it, we're determined to win it under his leadership and I'm looking forward to getting stuck into it."
Asked about claims he did not wholeheartedly support Mr Brown, he said: "No member of the government was involved in yesterday's letter, we're all determined to win the election under Gordon's leadership for the good of the country. I think it's very clear and we are all determined to see it through."
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