Mr Brown has expressed his "deep regret" over the issue
The government has defended its response to revelations that a Downing Street adviser sent e-mails discussing slurs against leading Conservatives.
Damian McBride resigned after unfounded claims about Tory leader David Cameron and others became known.
The prime minister's spokesman said he would not receive any severance pay.
Communities Secretary Hazel Blears said Gordon Brown had "taken action" by writing to those affected, but Tory sources have demanded a public apology.
The prime minister's spokesman said Mr Brown had been aware of the story on Friday but McBride had not resigned until Saturday because his boss had not known the exact nature of the e-mails until then.
The spokesman also said it had caused a "huge amount of frustration" in Downing Street that the controversy was diverting attention from efforts to deal with the real problems facing the country, such as the economy.
On Monday Mr Brown wrote to those mentioned in Mr McBride's e-mails, expressing his "deep regret".
David Cameron: 'Gordon Brown hired these people, he sets the tone'
A spokeswoman for Mr Cameron said the handwritten letters to those smeared showed the prime minister had "finally recognised the gravity of what's been happening in Downing Street".
In them, Mr Brown insisted no ministers had been involved in the e-mail conversation.
But BBC political correspondent Laura Kuenssberg said Conservative Party sources felt this action did not go far enough and was unlikely to reduce pressure on the prime minister.
Meanwhile, Frances Osborne, wife of shadow chancellor George Osborne and one of those mentioned in the e-mails, has complained to the Press Complaints Commission after allegations about her were repeated in the Sunday Times and News of the World.
The Conservatives have urged Cabinet Secretary Sir Gus O'Donnell to clarify who knew about the proposal to set up a Labour website, called Red Rag, where it was suggested the smears sent in Mr McBride's e-mails be published.
Mr McBride used his Downing Street account when he sent the e-mails in January to former government spin doctor Derek Draper.
Mr Draper runs the LabourList blog and was proposing to set up Red Rag, a gossip-led site. The idea was later abandoned.
However, the e-mails came to the attention of Paul Staines, author of the "anti-politics" Guido Fawkes blog, who revealed their existence. The row that followed forced Mr McBride's resignation.
As well as writing to those named in the e-mails on Monday, Mr Brown sent a letter to the cabinet secretary, saying he was ready to take "whatever action is required" to prevent a repeat of the incident.
He called for anyone caught "disseminating inappropriate material" to lose their jobs automatically, and suggested special advisers should not be allowed to use official resources for party political purposes.
Backbench Conservative MP Nadine Dorries, who was the focus of some of the e-mail correspondence, said the current code of conduct already included safeguards to prevent such behaviour.
People wanted to see this adhered to rather than "spinning new rhetoric about writing a new code", she said.
Mr McBride resigned after apologising for sending "juvenile and inappropriate" e-mails.
He had been moved to a strategy and planning role at 10 Downing Street last year, having previously dealt day-to-day with journalists.
This followed Mr McBride's handling of the news of Transport Secretary Ruth Kelly's departure during the Labour Party conference, which was seen by many party activists to have overshadowed the prime minister's speech.
Labour MP Paul Flynn told BBC Radio 4's Today programme: "This man has gone instantly. He did have a bit of form in the past.
"He was half-sacked before and exiled to a cupboard in the back of 10 Downing Street, I understand.
"But he's clearly someone who's behaved abominably and done a great deal of damage to the Labour Party."
Ms Blears told the BBC: "The e-mails that were sent by Damian McBride were utterly unacceptable. The prime minister's made that crystal clear.
If there remain people close to the prime minister who are thinking of fighting the forthcoming general election in a personal and dirty way then they should go
Labour MP Stephen Byers
"Damian McBride has gone through the door in pretty sharp order. The prime minister's written personally to the people involved, expressing great regret about what's happened, and he's toughening up the code for special advisers.
"Now that says to me that the prime minister, who knew nothing about these e-mails, has taken action on every single front here."
Labour MP Stephen Byers, a former cabinet minister, wrote in the London Evening Standard that he had been the victim of Mr McBride's "aggressive and hostile media briefing" several times.
"As a result I have to admit that I made little effort to suppress a smile when I heard about his enforced departure from Downing Street," wrote Mr Byers.
Nadine Dorries says Damian McBride was "inventing and fabricating stories"
"If there remain people close to the prime minister who are thinking of fighting the forthcoming general election in a personal and dirty way then they should go, and go now.
"This is the time for Mr Brown to demonstrate that as a leader he is focused on developing and implementing policies that will meet the challenges that face us as a country."
Fellow Labour MP Frank Field, a former minister, said: "Harold Wilson asserted that the Labour Party was a moral crusade or it was nothing.
"The McBride affair has left Labour members looking at nothing. That is the reality check that McBride has wrought on the party."
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