* Couple living in separate residences in Paris
* Separation occurred a month ago, source says
* Couple say will sue magazine for privacy invasion (Adds Strauss-Kahn and Sinclair decision to sue)
PARIS, June 29 (Reuters) - Former International Monetary Fund chief Dominique Strauss-Kahn, who is facing a probe into his alleged involvement with a prostitution ring in France, and his wife have separated, a source said.
Anne Sinclair, a wealthy heiress who recently relaunched her media career as a news editor at the Huffington Post's French edition, and Strauss-Kahn separated about a month ago and they are living in separate residences in Paris, said the source, who is close to Strauss-Kahn.
The weekly magazine Closer earlier reported in its online edition that Sinclair threw Strauss-Kahn out of their home in central Paris.
Lawyers for the couple said they intended to sue the magazine for invasion of privacy over the report, which also appeared on the front page of the print edition and in an article inside.
"Dominique Strauss-Kahn and Anne Sinclair have decided to sue this publication for invasion of privacy," lawyers Frederique Baulieu, Richard Malka and Henri Leclerc said in a statement.
Strauss-Kahn is under investigation in France to establish whether he knew he was dealing with prostitutes and pimps when he attended sex parties in northern France, Paris and Washington in 2010 and 2011 allegedly organized by business acquaintances.
Public prosecutors last month widened the inquiry to include a possible gang rape charge after a prostitute told them Strauss-Kahn and friends had forced her to have sex in a group when she came to Washington to meet him in December 2010. The woman has not filed a formal complaint.
Strauss-Kahn denies knowing that the women at the parties were prostitutes or that there was any violence.
His career at the head of the Washington-based IMF was cut short when he was arrested in New York in May 2011 on charges, which he denied and that have since been dropped, of attempting to rape a hotel maid. After criminal charges were abandoned over concerns about her credibility, the maid, Nafissatou Diallo, pressed ahead with a civil case.
Strauss-Kahn, a former French finance minister, and Sinclair have been married for 20 years. She stuck close by him when he faced the rape accusations in New York.
The scandal destroyed his hopes of running for French president in the April-May 2012 election for the Socialist Party, which instead won power under Francois Hollande.
News of the separation comes after weeks of media speculation that the relationship was under pressure, in part as Strauss-Kahn grew depressed at his lack of career options.
"He's in a bad way. It's very sad," a person who knows Strauss-Kahn and recently saw him socially told Reuters this month. "He's mostly just at home on his own while Anne is out and about with her new job. He's shunned by everybody." (Reporting By Catherine Bremer, additional reporting by Joseph Ax in Washington and Nicholas Vinocur in Paris; Editing by Michael Roddy)