Russia has resisted calls to return the so-called Schneerson collection to the New York-based Hasidic Chabad-Lubavitch group, descendants of the last private owner of the writings, and Putin said they were part of Russia's cultural heritage.
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"For the Jewish people, Russia has been a homeland for centuries, as it remains so today," Putin said while visiting the museum to launch its latest exhibition.
"I hope that moving the Schneerson library to the Jewish Museum and Tolerance Center...will put an end to this problem once and for all," said the former KGB spy, who has sought to celebrate Russia as country of many religions while fostering close ties with the Russian Orthodox Church.
The Schneerson collection consists of thousands of Jewish books, religious papers and manuscripts, some of them dating back to the 16th century, their leather-covered spines showing the effects of age.
The 4,425 books that will be kept at the museum include editions of the Torah and Talmud with unique margin notes by Hasidic leaders of the Chabad-Lubavitch community, which considers the whole collection its inheritance.
They were left for safekeeping from the turmoil of the World War I in a warehouse in what is now western Russia's Smolensk province, but later were taken by the newly installed Bolshevik state and finally kept in Russia's state library until recently.
"Jewish books should be held in Jewish organizations," said Alexander Boroda, head of the Russian Federation of Jewish Communities and the museum's director.
"It is a restoration of historical justice that they will be managed by the Jewish community."
The 500 books that have been brought to the museum are held in glass-covered bookstalls in a room with regulated humidity and temperature set at 18 degrees C (64 F) to preserve the paper. The rest are to be moved there by the end of the year.
It remains unclear whether the move will defuse the diplomatic and legal tug-of-war that started even before the 1991 collapse of the Soviet Union and has weighed on bilateral ties between Moscow and Washington.
Moscow reacted angrily when a US judge ruled in January that Russia should pay $50,000 a day in fines for failure to return the books.
Chabad-Lubavitch declined to comment on the matter, while the US ambassador to Moscow said talks were still ongoing.
"We continue to work with all sides - and there are many sides in this discussion – on a resolution that will be acceptable to all sides, and irrespective of what happened today we continue to do that," Ambassador Michael McFaul said.
Another part of the Schneerson collection rests in Russia's military archive after being confiscated by Soviet troops in Nazi Germany during World War II.
Those papers had fallen into Nazi hands after their last private owner, the late Chabad-Lubavitch Rabbi Yosef Schneerson, fled the Soviet Union in late 1920s and wandered around eastern Europe in search of a safe place.
The Chabad-Lubavitch community originated under the Russian Empire and Yosef Schneerson was born to it in 1880. He set up the collection to bring together religious books and writings of his kin before fleeing for New York where he died in 1950.
Up to one million Jews live in Russia after the population dwindled in tsarist-era pogroms, Soviet oppression of religion and emigration in recent decades.