But if a better deal wasn't available and people were forced to choose, 41% said Britain should go ahead and leave on those terms, 32% think there should be a referendum on whether to stay in the European Union after all, 27% of people are unsure.
Britain voted 52 percent to 48 percent to leave the European Union in a referendum on June 23.
For Leavers, the research suggests that more people would want to stick with the country's decision and leave even if Brexit negotiations are disappointing.
Tusk reiterated on Monday that he would send governments a draft of Brexit negotiating guidelines within 48 hours of May's letter, which will set out Britain's demands for the talks.
Mr Tusk has previously said he expects to call an extraordinary summit of the 27 other members within four to six weeks, to draw up a mandate for the European Commission's chief negotiator, Michel Barnier.
Downing Street said she would write a letter to the European Council, adding that it hoped negotiations on the terms of exit and future relations could then begin as quickly as possible.
And it's not even certain that the United Kingdom - made up of England, Scotland, Wales and Northern Ireland - will survive the European Union exit intact.
Britain's ambassador to the EU, Tim Barrow, will also play a major role, and the Foreign Office will talk to individual member states to try to get them on its side.
The Prime Minister is expected to trigger Article 50 within the next fortnight - before her self-imposed deadline of the end of March. We have a plan for Britain and we also have a very clear plan for Brexit. Once May and her administration do this next week, however, the United Kingdom will have up to two years to settle any and all details of the Brexit process before the separation is complete.
However, Nicola Sturgeon's demand for a second independence referendum in Scotland last week - coupled with the delay in getting the Article 50 bill through the House of Lords - limited the government's choices. The EU wants Britain to pay a hefty divorce bill - estimates have ranged up to 60 billion euros ($64 billion) - to cover pension liabilities for EU staff and other commitments the United Kingdom has agreed to. May has said now is not the time for Scotland to vote on independence, preferring to wait until after Brexit.
Negotiations will also soon hit a fundamental topic: Britain wants "frictionless" free trade, but says it will restore controls over immigration, ending the right of European Union citizens to live and work in Britain. Britain hasn't ruled out a payment, but is sure to quibble over the size of the tab.