The Australian publisher and activist, 50, has been held in Belmarsh Prison in London since 2019 after the United States took legal action to extradite him.
The couple, who met when Assange was living in Ecuador's London embassy, have two children.
"I am relieved that reason prevailed and I hope there will be no further interference with our marriage," Ms Moris told The Press Association.
The couple have been engaged for a number of years and have been trying to get married despite ongoing legal action.
Their sons Gabriel, four, and two-year-old Max are British citizens.
The couple were taking legal action against the prison governor and Justice Secretary Dominic Raab, accusing them of preventing a wedding being held.
The application to wed was "received, considered and processed in the usual way by the prison governor, as for any other prisoner" a prison service official said.
Assange was arrested by British police after spending five years living in the Ecuadorian embassy in London, where he had sought political asylum as he fought to avoid extradition to Sweden over a separate case that was dropped in November 2019.
He feared he would be taken to the US for questioning over the activities of WikiLeaks.
In April 2019 has was jailed for 12 months for skipping bail but was kept in Belmarsh while a lengthy legal case was mounted by the US.
In January a judge refused the US's request to extradite Assange, but an appeal was lodged, with the outcome still pending.
No date has been set for the wedding.