Gay rights groups, which suffered the stinging defeat of a same-sex marriage bill in New York State in 2009, will publicly mount a new campaign for the legislation starting this week, relying on the popular Democratic governor, Andrew M. Cuomo, to overcome Republican resistance and their own history of poor coordination.

Under the supervision of the governor's staff, the groups intend to raise more than $1 million for a media blitz, hire a powerful political consultant close to the Cuomo administration and deploy field organizers to the districts of more than a dozen key lawmakers to drum up support, according to interviews with those involved in the effort.