Jo Brand will not face police action over a remark she made on the radio about throwing battery acid over politicians.
Scotland Yard were considering an allegation of incitement to violence.
Jo Brand made the comments on Radio 4 programme Heresy on Tuesday and has since apologised, describing them as “crass and ill-judged”.
The remarks sparked an outcry, with Brexit Party leader Nigel Farage among the critics, and the Met said on Thursday it was assessing them following an allegation of incitement to violence.
On Friday the force said: “Police received an allegation of incitement to violence on 13 June, relating to comments made on a radio programme.
“The referral has been considered by the MPS and no further police action will be taken in relation to this allegation.”
In reply to a question about the state of UK politics, Brand had said: “Well, yes, I would say that but that’s because certain unpleasant characters are being thrown to the fore and they’re very, very easy to hate and I’m kind of thinking ‘Why bother with a milkshake when you could get some battery acid?’
“That’s just me. I’m not going to do it, it’s purely a fantasy, but I think milkshakes are pathetic, I honestly do, sorry.”
Comedian David Baddiel, who created Heresy, said the BBC was wrong to edit the joke out of a repeat of the programme.
Speaking to Newsnight, he said: “I don’t think I would have nipped it out. Morally wrong? I’m not sure. I think they’re just trying not to cause trouble.
“The BBC are still to some extent the aunty of the nation and they don’t like trouble. Even though they did commission a show, Heresy, that was designed to push the boundaries of what people might think and say.
“If it was up to me, I would have kept that line in for the repeat. Apart from anything, it’s a bit silly when it’s had massive coverage to cut it out – that looks a bit cowardly.”
Brand has since apologised for the joke, describing it as “crass and ill-judged”.
The BBC said it regretted any offence caused by the radio programme, which was never intended “to encourage or condone violence”.