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The Survation poll for ITV television was conducted on Friday and Saturday, before an attack in London by Islamist militants that killed seven people and injured 48.
In the previous Survation poll for ITV, published a weekago, the Conservatives had a lead of six points.
But a separate Survation poll, published on Saturday for the Mail on Sunday newspaper, also gave the Conservatives a one-point lead.
Survation said its latest poll put support for Mrs May's Conservative Party at 41.5% compared with 40.4% for Labour, a result which if replicated in the election would put in jeopardy the Conservatives' majority in parliament.
Opinion polls by other leading polling firms have given wider leads for the Conservatives in recent days, ranging as high as 11 and 12 points.
Monday's Survation/ITV poll was based on interviews with 1,103 people.
Three weeks ago, a series of surveys showed Mrs May was on course for a landslide parliamentary majority which she called to secure a strong mandate for Brexit talks.
But Mrs May's campaign began to struggle after she proposed a plan to make elderly people pay for more of their social care, even though she moved quickly to say there would be a limit on the amount of costs that people would face.
It remains unclear whether Saturday's attack will have an impact on the election.
The Conservatives' lead over Labour continued to fall after a suicide bombing in Manchester on 22 May but polling firms have linked the narrowing to the rival parties' policy proposals rather than the attack.
50% of respondents in Monday's poll thought Mrs May would make a better prime minister than Labour leader Jeremy Corbyn but his credibility as a potential leader of the country has risen to 36% from 15% in early May.
Several other polls are due to published before voters go to the polls on Thursday.