Lottery process that selects the 764 gay couples and heterosexual couples can marry on that day.
"The number of couples who want to get married on Sunday exceeded the amount that can be handled by the civil registry offices of the city," Mayor Michael Bloomberg.
On 24 June, the state of New York became the largest U.S. state that allows same-sex marriage.
There is now six U.S. states plus the district of Washington DC that allow gay marriage.
In eight other states allowed gay couples to register their relationship status in a civil ceremony, not as a marriage, but their equal rights with married couples.
Sweepstakes registration of marriage in New York for this Sunday's open Tuesday afternoon New York time (Wednesday, 20.07 midnight GMT) and closed at the same time on Thursday (07/21/2011).
Colin Cunliffe and Brewter McCall is among gay couples who celebrate the new marriage law
The lottery winners will be notified of a marriage license on Friday.
"We will make history on Sunday, and the eyes of this country will once again be directed to the City of New York," Bloomberg said.
"The thing that we do not want is a couple who want to get married should be in line for hours, and ended up going with the upset in the day that should be the happiest day of their lives," he added.
"The most fair way to determine who is entitled to get married on Sunday and make sure everyone has time to plan a wedding day the most important to them is through a lottery system."
Since July 5 there are 2661 couples who register to marry on Sunday and 1728 pairs of them are same-sex couples, the government announced the City of New York.