A stock valuation released by one of the Russian banks involved in the issue has priced the stake at about USD 1.4-USD 1.6 billion (1.0-1.2 billion euros).
Alrosa expects "that the offering will expand our access to the international capital markets and help to unlock the value of the company," said company chief executive Fyodor Andreyev.
The gem producer had remained one of Russia's most secretive companies despite its global ambitions and long history of providing the federal government with budget revenues at times of economic malaise.
Alrosa had said in 2011 that its existing mines would provide it with diamonds for the coming four decades only.
Russia remains a tempting but unpredictable market for foreign investors who are interested in tapping its enormous mineral wealth.
Foreign direct investment has been slowed by a perceived lack of court independence and weak ownership laws that turn investing in Russia into an unduly high risk.
President Vladimir Putin appeared to acknowledge the problem during a speech at a business forum in Moscow in which he pledged once again to pursue overdue structural reform.
"We will definitely keep our focus on development and the creation of a strong economy."
Russian media said a fund linked to the founder of the Anglo-American diamonds miner and trader De Beers as well as some top Chinese investors had already expressed interest in the Alrosa shares.
The east Siberian firm accounts for roughly a quarter of the world's diamond production and has reported reserves of 1.28 billion carats.
But the company had been riddled with debt and criticised by analysts for opaque accounting practises.
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