LONDON—The British government promised new powers for Scotland in areas including taxes, spending and welfare if it votes to remain in the U.K. after a weekend opinion poll showed a surprising surge in support for Scottish independence.

A poll released late Saturday showed pro-independence voters in the lead for the first time since the Scottish referendum campaign began. The shift in sentiment ahead of the Sept. 18 vote raised the prospect that a split that seemed unlikely a few weeks ago is now thinkable. The increase in...