The Institute for Government survey found that figure dropped to just a quarter (25%) for Remain voters in a sign of the challenge facing Theresa May.
The Prime Minister has given no clear signal on whether the country intends to stay in the European single market or not and membership is seen by EU leaders as incompatible with getting full control over immigration.
The conundrum has become central to the Brexit debate and has caused friction within the Government as ministers weigh up how to maintain the benefits of the trading bloc while regaining full control of the UK’s borders - a goal deemed by Mrs May as essential to properly implement the referendum result.
Meanwhile, the poll also found rising levels of trust in politics, with 8% more people than in 2014 believing politicians prioritise policies that are best for the country.
Similarly, 7% more believe they are prioritising long-term decisions and 5% more think they are prioritising running government professionally.
Respondents also thought politicians were less likely to focus on scoring political points (down 4%) or getting re-elected (down 8%) than in 2014.
Populus interviewed 2,035 British adults online between August 31 and September 1. Data is weighted to be nationally representative.