Assange declined to give Snowden's whereabouts because of "threats" from the United States.
Snowden, the target of a US arrest warrant over the leaking of massive surveillance programmes, has vanished in Moscow after failing to take a flight to Cuba from where he had been expected to continue to Ecuador and claim asylum.
Assange said Snowden and the WikiLeaks staff member who had travelled with him to Moscow from Hong Kong on Sunday, Sarah Harrison, are "healthy and safe and they are in contact with their legal team".
"I cannot give further information as to their whereabouts or present circumstances other than to say that the matter is in hand," Assange added in a conference call with journalists from Ecuador's embassy in London.
"We are aware where Mr Snowden is, he is in a safe place and his spirits are high.
"Due to the bellicose threats coming from the US administration we cannot go into further detail at this time. Unfortunately we cannot reveal what country he is in at this time."
Asylum
Assange said Snowden had left his bolthole in Hong Kong on Sunday with a refugee document supplied by Ecuador after the United States revoked his passport.
"To leave Hong Kong Mr Snowden was supplied with a refugee document of passage by the Ecuadoran government," he said, adding that the asylum process was not complete.
Ecuador has been harbouring Assange in its London embassy for a year after he fled there to avoid deportation to Sweden where investigators want to question him about accusations of rape.
Assange said Snowden left Hong Kong on Sunday "bound for Ecuador via a safe passage through Russia and other states".
Ecuador is still considering the asylum application.
He said Snowden had had "no advance communication" with Russian officials before he had headed for Moscow, and he described allegations that the American had been debriefed by Chinese officials in Hong Kong as "false".
'Traitor'
​Apart from Ecuador, Snowden has made asylum applications to several other countries including Iceland, WikiLeaks spokesperson Kristinn Hrafnsson – who himself is Icelandic – said in the conference call, without naming the other nations.
US Secretary of State John Kerry has dubbed Snowden a traitor to his country and warned both Russia and China that their relations with the United States might be damaged by their refusal to extradite him.
Assange claimed that the "kidnapping or incapacitating or Mr Snowden must have been considered" by the United States if it perceived that doing so "would terminate the exposure of their spy programme".
"So I believe Mr Snowden was well advised to go public at the time to protect his safety and the safety of the journalists involved," the Australian said. – AFP