PERTH, Scotland (Reuters) - The Scottish National Party will push for a new referendum on independence as an alternative to Brexit, party leader Nicola Sturgeon said on Tuesday, as the SNP digs its heels in on an issue that may cost her party support in Britain’s June 8 election.

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“There is just too much at stake for Brexit simply to be imposed on Scotland, no matter how damaging it turns out to be,” Sturgeon, who is also first minister of the devolved Scottish government, told a launch of the SNP’s policy offer to voters.

British Prime Minister Theresa May has rejected the SNP’s demand for a new choice on secession, saying that with Britain about to leave the European Union this is not the time for another vote, without saying when that time might be.

Sturgeon wants a new choice to be offered when it is clear what Brexit means.

The SNP would ensure that Scotland’s voice was heard, Sturgeon said, and the party would push for Scotland to stay in the single market even as Britain leaves it. But SNP proposals along those lines, presented in December, were rejected by May’s government.

The SNP’s policy document said there was no “rational argument” for leaving the single market.

Scots rejected independence by a 10-point margin in 2014, and polls show support has not shifted since then. But Sturgeon says a new choice should be offered because Scottish voters voted last year to remain in the EU, while the UK as a whole voted to leave

The SNP is the dominant party representing Scotland in the British parliament, with 54 of 59 seats. It also runs the devolved Scottish government, which has authority over health, education and transport and some portion of taxes.

Polls appear to show that while the SNP will easily win the election in Scotland, the Conservatives will take some seats from them by sticking to practically one policy - opposing independence.

The Conservatives says that the SNP’s answer to any issue is always independence, and they are using Brexit as an excuse to try to further their cause.

“Strip away the bluster and it’s written down in black and white - (Sturgeon) wants to drag Scotland back to another referendum by as early as next Autumn,” Scottish Conservatives leader Ruth Davidson said in a statement..

The SNP argues that Brexit is a game-changer.

“At the end of the Brexit process - not now, but when the terms of the deal are known - Scotland must have a choice about our future; a choice between following the UK down the Brexit path or becoming an independent country,” Sturgeon said, adding that the Scottish parliament had already backed the devolved government to have that choice.

She said the party sought to represent even those who voted to leave the EU, but it is concerned about the “extreme Brexit” being pursued by the Conservatives.