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The turnabout by Mr. Cuomo, who had long resisted legalizing medical marijuana, comes as other states are taking increasingly liberal positions on it — most notably Colorado, where thousands have flocked to buy the drug for recreational use since it became legal on Jan. 1.

Mr. Cuomo's plan will be far more restrictive than the laws in Colorado or California, where medical marijuana is available to people with conditions as mild as backaches. It will allow just 20 hospitals across the state to prescribe marijuana to patients with cancer, glaucoma or other diseases that meet standards to be set by the New York State Department of Health.

While Mr. Cuomo's measure falls well short of full legalization, it nonetheless moves New York, long one of the nation's most punitive states for those caught using or dealing drugs, a significant step closer to policies being embraced by marijuana advocates and lawmakers elsewhere.

New York hopes to have the infrastructure in place this year to begin dispensing medical marijuana, although it is too soon to say when it will actually be available to patients.

Mr. Cuomo's change of heart comes at an interesting political juncture. In neighboring New Jersey, led by Gov. Chris Christie, a Republican whose presidential prospects are talked about even more often than Mr. Cuomo's, medical marijuana was approved by his predecessor, Jon S. Corzine, a Democrat, but implemented only after Mr. Christie put in place rules limiting its strength, banning home delivery, and requiring patients to show they have exhausted conventional treatments. The first of six planned dispensaries has already opened.

Meanwhile, New York City's new mayor, Bill de Blasio, had quickly seemed to overshadow Mr. Cuomo as the state's leading progressive politician.

For Mr. Cuomo, a Democrat who has often found common ground with Republicans on fiscal issues, the sudden shift on marijuana — which he will announce on Wednesday in his annual State of the State address — was the latest of several instances in which he has embarked on a major social policy effort sure to bolster his popularity with a large portion of his political base.

In 2011, he successfully championed the legalization of same-sex marriage in New York. And a year ago, in the aftermath of the mass school shooting in Newtown, Conn., Mr. Cuomo pushed through legislation giving New York some of the nation's toughest gun-control laws, including a strict ban on assault weapons. He also has pushed, unsuccessfully so far, to strengthen abortion rights in state law.

The governor's action also comes as advocates for changing drug laws have stepped up criticism of New York City's stringent enforcement of marijuana laws, which resulted in nearly 450,000 misdemeanor charges between 2002 and 2012, according to the Drug Policy Alliance, which advocates more liberal drug laws.

During that same period, medical marijuana became increasingly widespread outside New York, with some 20 states and the District of Columbia now allowing its use.

Mr. Cuomo voiced support for changing drug laws as recently as the 2013 legislative session, when he backed an initiative to decriminalize so-called open view possession of 15 grams or less. And though he said he remained opposed to medical marijuana, he indicated as late as April that he was keeping an open mind.

His about-face, according to a person briefed on the governor's views but not authorized to speak on the record, was rooted in his belief that the program he has drawn up can help those in need, while limiting the potential for abuse. Given Mr. Cuomo's long-held concerns, this person said, he insisted that it be a test program so he can monitor its impact.

But Mr. Cuomo is also up for election this year, and polls have shown overwhelming support for medical marijuana in New York: 82 percent of New York voters approved of the idea in a survey by Siena College last May.