MPs will get a vote on whether to hold another referendum if they back the EU Withdrawal Agreement Bill, she said.
The bill also contains new guarantees on workers' rights, environmental protections and the Irish backstop as well a customs "compromise".
If MPs reject the bill, she warned them a negotiated exit would be "dead in the water" and Brexit could be stopped.
MPs have rejected the withdrawal agreement negotiated with the EU three times.
In what is seen as a last roll of the dice, Mrs May is now bringing the Withdrawal Agreement Bill - legislation required to bring the agreement into UK law - to Parliament in early June.
In a speech in London, the prime minister said the deadlock over Brexit was having a "corrosive" impact on British politics and stopping progress in other areas.
"We are making a new offer to find common ground in Parliament," she said. "That is the only way to deliver Brexit."
She said MPs would have the chance to vote on whether to hold another referendum before the exit agreement was ratified and, if MPs backed it, the government would make the necessary arrangements.
While she personally opposed another referendum on the terms of Brexit, the PM said she recognised the "genuine and sincere" feelings on the issue in Parliament,
She urged MPs to back the Withdrawal Agreement Bill at its first parliamentary hurdle in and then "make the case" for another public vote when the bill was examined in detail.
Mrs May also said there would be new legislation to ensure UK workers' rights were "no less favourable" than within the EU and guarantees that there would no dilution in environmental standards.
In a move to reassure Tory Brexiteers, she said there would be a legal obligation on the government to find alternative arrangements to maintain an open border in Northern Ireland by the end of 2020 to ensure the controversial backstop plan never had to come into force.
Northern Ireland would remain aligned with the rest of the UK, in terms of regulation, after Brexit and would not be part of a separate customs territory.
Appealing to MPs to back her plan, she said it would honour the 2016 referendum result, adding: "I have compromised, now I ask you to compromise too."