It showed that a clear majority want a General Election at the earliest possible opportunity (62%), compared with 36% who disagreed.
The same poll asked if David Cameron has what it takes to be a good Prime Minister - 50% agreed compared with 41% who disagreed.
The results were released on Monday's show by presenter Anita Anand, and discussed with Labour backbenchers, Kevin Barron, chair of the Health Select Committee, and the former Minister Mark Fisher.
Changing the leader will not change the result, poll finds
The poll was conducted for BBC2's Daily Politics programme by ComRes.
It spoke to 1,001 voters between 5 and 7 June 2009 - after polling day, but before the European results were released overnight on Sunday.
Full Results:
Do you agree or disagree with the following statements:
1) Gordon Brown has lost authority and should resign immediately Agree 52% Disagree 43% Refused/Don't know 5%
2) A General Election should be called at the earliest possible opportunity: Agree 62% Disagree 36% Refused/Don't know 2%
3) Labour could win the next election under a different leader: Agree 32% Disagree 63% Refused/Don't know 6%
4) David Cameron has what it takes to be a good PM: Agree 50% Disagree 41% Refused/Don't know 10%
Bookmark with:
What are these?