Anatoly Kucherena said that Snowden's whereabouts will be kept secret for security reasons. The former NSA systems analyst was stuck at Moscow's Sheremetyevo airport since his arrival from Hong Kong on June 23.
"He now is one of the most sought after men in the world," Kucherena told reporters at the airport. "The issue of security is very important for him."
The U.S. has demanded that Russia send Snowden home to face prosecution for espionage, but President Vladimir Putin dismissed the request.
Putin had said that Snowden could receive asylum in Russia on condition he stops leaking U.S. secrets. Kucherena has said Snowden accepted the condition.
The Guardian newspaper on Wednesday published a new report on U.S. intelligence-gathering based on information from Snowden, but Kucherena said the material was provided before Snowden promised to stop leaking.
Snowden, who revealed details of a U.S. intelligence program to monitor Internet activity, has received offers of asylum from Venezuela, Nicaragua and Bolivia and said he would like to visit those countries. However, the logistics of reaching any of those countries are complicated because his U.S. passport has been revoked.
Snowden's father said in remarks broadcast Wednesday on Russian television that he would like to visit his son. Kucherena said he is arranging the trip.
The lawyer has said earlier that the temporary asylum would allow Snowden to travel freely around Russia, but wouldn't allow him to leave the country. The one-year asylum can be extended.
WikiLeaks, a group which has adopted Snowden's cause, said its legal adviser Sarah Harrison is now with him. The group also praised Russia for providing him shelter.
"We would like to thank the Russian people and all those others who have helped to protect Mr. Snowden," WikiLeaks said on Twitter. "We have won the battle -- now the war."
Kucherena said that Snowden spent little time packing and left the airport in a taxi. The lawyer said the fugitive had friends in Russia, including some Americans, who could help ensure his security, but wouldn't elaborate.
Snowden's case has further strained U.S.-Russian ties already tense amid differences over Syria, U.S. criticism of Russia's human rights record and other issues.
Putin's foreign affairs aide, Yuri Ushakov, sought Thursday to downplay the impact this will have on the relations between the two countries.
"This issue isn't significant enough to have an impact on political relations," he said in remarks carried by Russian news agencies.
He said that the Kremlin hasn't heard any signal from Washington that Obama could cancel his visit to Moscow ahead of next month's G-20 summit in St.Petersburg.
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