He feared Sweden would hand him over to the United States to face prosecution over WikiLeaks' publication in 2010 of thousands of classified military documents and diplomatic cables in one of the largest information leaks in U.S. history.
Deputy Labor leader Tanya Plibersek, speaking in Sydney earlier, said Assange still had questions to answer, but Mr Barns said there were no outstanding issues and he would welcome a call from her.
Assange has been holed up in the Ecuadorean Embassy in London since 2012 to avoid extradition to Sweden over the allegations.
"This is a total victory for us", Assange's lawyer Per Samuelson said.
Swedish prosecutors interviewed Assange at the embassy last November and received a full translation of the interview in mid-March, which they have since been reviewing.
Ny says she has withdrawn a European arrest warrant.
However, Assange is still a "wanted man" in the United Kingdom for not showing up to a court date, which the Metropolitan police noted to AP is a "much less serious offense" than sex crimes and said they "will provide a level of resourcing which is proportionate to that offense".
The Metropolitan Police, however, swiftly confirmed it would still arrest the Australian over the lesser charge of skipping bail in 2012 when he sought asylum.
Assange had disputed the allegations and argued that he risked being extradited from Sweden to the United States and tried for espionage.
The probe has been dropped because there was "no reason to believe that the decision to surrender him to Sweden can be executed in the foreseeable future", said Swedish prosecutors. So far, the only reaction from Assange has been a tweeted picture, showing Assange smiling proudly. The South American country has granted him asylum.
Swedish director of public prosecutions Margaret Ny said the decision to drop the rape investigation had been made after all possibilities for charges had been "exhausted".
Assange has disputed the rape allegations.
David Allen Green, a law commentator for The Financial Times, suggested that if Assange left the embassy and was arrested, Sweden could resume its case.
A lawyer for the woman who alleges she was raped by Julian Assange says "it is a scandal that a suspected rapist can avoid the judicial system and thus avoid a trial in court".
Assange had been remanded in custody by Sweden "in absentia" over a 2010 rape allegation, and has been taking refuge inside Ecuador's embassy in London since 2012 in order to escape the warrant, citing fears he may be extradited to the U.S. to be tried over WikiLeaks' publication of thousands of classified documents.
The police ended their round-the-clock guard outside the embassy in October 2015 but said they were strengthening a "covert plan" to prevent his departure. He originally sought refuge there, due to fears that once in Sweden, he might be extradited to the United States where he is wanted for leaking classified military and diplomatic documents.