LONDON, June 6 (Reuters) - Sterling fell 1 percent against the dollar to hit a three-week low on Monday, while the cost of hedging against swings over the coming month traded at its highest since early 2009 on growing concerns over whether Britain will stay in the European Union.

Sterling has been weighed down since late last year by worries that the June 23 referendum on EU membership could lead to Britain leaving the bloc.

Britain’s hefty current account deficit - 7 percent of output in the last quarter of 2015 - makes the economy, and the currency, vulnerable to any pull-back in investment flows.

Polling firm TNS said on Monday that the campaign to get Britain out of the European Union had a 2-point lead over the “Remain” campaign, while a YouGov poll for ITV showed that the “Leave” campaign had a 4 point lead.

Bookmakers shortened their odds on Brexit in response, with betting website Betfair putting the chances of a vote to leave at around 30 percent on Monday. The odds were at around 27 percent at the start of last week.

Against the dollar, sterling was 1 percent lower at $1.4370, having fallen to $1.4352 in early Asian trading, its lowest in three weeks. The euro was 0.75 percent higher at 78.95 pence.

“The polls are likely to make people rather uneasy and we can see that quite clearly today in the pound,” said Craig Erlam, senior market analyst at OANDA.

“With both sides likely to step up their game over the next couple of weeks, I imagine we’ll see a lot more volatility in the pound and the closer the polls get, or if ‘Vote Leave’ continues to push ahead, the pound may find itself back towards April’s lows before too long.”

Reflecting the nervousness, the one-month sterling/dollar implied volatility -- a gauge of how sharp swings will be over the June 23 referendum date -- traded at 21.4 percent, its highest level since early 2009.

For a graphic of opinion polls: tmsnrt.rs/1Ke31HF (Reporting by Anirban Nag; Editing by Alison Williams)