The Last Ship, a story of unemployed shipbuilders who take over a factory to build a new vessel, opened on 26 October to mixed reviews.
Last month, Sting stepped in to replace leading man Jimmy Nail in a bid to revive flagging ticket sales.
His appearance had a "galvanizing impact", but producers felt attendance would drop when he left, especially during the slow winter period.
News of the closure came after Broadway theatres reported a record-breaking year for attendance and box-office takings in 2014.
A total of 13.1 million people went to see shows in the heart of New York last year, bringing in $1.36bn (Â£0.89bn).
The Broadway League, which represents producers and theatre owners, said attendance was up 13 percent and box-office takings up 14 percent.
The week ending Sunday 28 December was the biggest Christmas week in history for the theatre district - but the best-attended and highest-grossing period overall was New Year week.
Long-running shows Wicked, The Lion King and The Book of Mormon were among the top-grossing shows, with each pulling in more than $2.2 million (Â£1.45 million) for the New Year week.
Nearly 20 new shows are scheduled to open on Broadway in 2015, including a musical version of the 2003 Jack Black film School of Rock.
It is written by Downton Abbey creator Julian Fellowes, with music by Andrew Lloyd Webber.
Sting's Broadway musical is to close on 24 January, after just three months.