Assange, who denies the accusation, has avoided extradition to Sweden for seven years after seeking refuge at the Ecuadorean embassy in London in 2012.
The 48-year-old Australian was evicted in April and sentenced to 50 weeks in jail for breaching his bail conditions.
He is currently being held at Belmarsh prison in London.
The Swedish investigation had been shelved in 2017 but was re-opened earlier this year following his eviction from the embassy.
Separately, the US is seeking Assange's extradition from the UK over his alleged role in the release of classified military and diplomatic material by Wikileaks in 2010.
Deputy Director of Public Prosecution Eva-Marie Persson took the decision to "discontinue the investigation regarding Julian Assange", the Swedish Prosecution Authority said.
"The reason for this decision is that the evidence has weakened considerably due to the long period of time that has elapsed since the events in question," it added.
Ms Persson said: "I would like to emphasise that the injured party has submitted a credible and reliable version of events.
"Her statements have been coherent, extensive and detailed; however, my overall assessment is that the evidential situation has been weakened to such an extent that that there is no longer any reason to continue the investigation."
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