Dame Margaret Hodge, a senior Jewish MP, said that Corbyn cannot resolve the issue of anti-Semitism because he "is the problem".
At least three Labour MPs are considering following veteran MP Frank Field and resigning the party whip if the NEC fails to adopt the full definition of anti-Semitism. John McDonnell, the shadow chancellor, has indicated that Labour could adopt the international definition then decide on controversial caveats at a later date in an effort to defuse the row.
Brown told the conference: "This is not a theoretical and abstract problem we are dealing with. It is about the soul of the Labour Party. We cannot stand up for the rights of some of the people, some of the time. We have to stand up for all of the people all of the time.
"This stain has got to be removed. The hurt it has done has got to be undone immediately and within the next few days."
He warned that amending the international definition would "destroy the unity" that is essential to fighting anti-Semitism. "The Labour Party has been the last line of defence for people facing persecution and discrimination. The last line of defence must never become the front line of the problem."
Lord Sacks, a former chief rabbi, last week accused Corbyn of being anti-Semitic and compared comments he made before he was leader to Enoch Powell's "rivers of blood" speech.
Yesterday, he told the BBC: "Jews have been in Britain since 1656, I know of no other occasion in these 362 years where the majority of our community are asking is this country safe to bring up our children. It is very very worrying. When people hear the kind of language coming out of Labour, that has been brought to the surface in Jeremy Corbyn's earlier speeches, they cannot but feel an existential threat."
Momentum, the grassroots group that backs Corbyn, was today to call for all Labour MPs to face mandatory reselections ahead of a general election. The plans will be seen as an effort to purge moderate MPs from the party.
Watson said he launched No Space for Hate because he was "increasingly disappointed at the tone of political discourse on social media".