The Wikileaks founder and Ms Moris have two sons together, who she said were conceived while he was living inside London's Ecuadorean embassy.
The prison service said Mr Assange's application was "considered in the usual way by the prison governor".
Ms Moris told PA she was "relieved that reason [had] prevailed".
She added: "I hope there will be no further interference with our marriage."
Inmates are entitled to apply to be married in prison under the Marriages Act 1983 and where applications are granted, they must meet the full costs of the marriage, with no taxpayer help.
In an interview with the Mail on Sunday last year, Ms Moris, a South African-born lawyer, revealed that she had been in a relationship with Mr Assange since 2015 and had been raising their two young sons on her own.
In a video posted on Wikileaks' YouTube account, she said she had met Mr Assange in 2011 when she joined his legal team.
Ms Moris added that she had visited him in the embassy almost every day and had "got to know Julian very well".
The couple fell in love in 2015 and got engaged two years later.
Ms Moris said that Mr Assange had watched both boys being born via video link and they had visited their father at the embassy.
Mr Assange, 50, continues to fight extradition to the US on espionage charges.
He is wanted in the US on allegations of conspiracy to obtain and disclose national defence information, following Wikileaks' publication of hundreds of thousands of leaked documents relating to the Afghanistan and Iraq wars.
The publications include the release in April 2010 of footage showing US soldiers shooting and killing civilians from a helicopter in Iraq.
The Australian has been in Belmarsh Prison since 2019, when he was carried out of the Ecuadorian embassy in London by police and arrested for breaching his bail conditions.
He had been in the embassy since 2012, avoiding extradition to Sweden, where he faced sex offence allegations. He has always denied those and they were eventually dropped.
A Prison Service spokesperson said: "Mr Assange's application was received, considered and processed in the usual way by the prison governor, as for any other prisoner."