New York voters on Tuesday approved a constitutional amendment to expand casino gambling, authorizing as many as seven full-scale casinos as part of a plan meant to bring jobs to economically distressed upstate regions.

The proposal, which was championed by Gov. Andrew M. Cuomo, was supported by 57 percent of voters, with most precincts reporting.

The approval is a milestone in the gambling industry’s long, expensive push to tap into the New York market, an effort that has spanned decades, cost tens of millions of dollars and is certain to continue as gambling companies vie for the right to develop the new casinos.

New York State already has five Indian-run casinos, all of them upstate, and nine slot machine parlors at racetracks. And the State Legislature, at the urging of Mr. Cuomo, has required that at first, only four new casinos will be permitted, and only upstate: in the Albany area, the Catskills-Hudson Valley region and part of the Southern Tier, a region along the northern border of Pennsylvania.