Assange is wanted by US authorities to face trial on 18 counts relating to WikiLeaks' release of vast troves of confidential US military records and diplomatic cables.
The 50-year-old, who denies any wrongdoing, has been held at Belmarsh prison since 2019 and before that was holed up in the Ecuadorean embassy in London for seven years.
While living at the embassy he fathered two children with Moris, a lawyer who he met in 2011 when she started work on his legal team. Their relationship began in 2015.
Wednesday's registrar-led ceremony will take place during visiting hours at the prison, where some of Britain's most notorious criminals have served sentences.
Moris's wedding dress and Assange's kilt - a nod to his family ties to Scotland - have been created by British fashion designer Vivienne Westwood, who has previously campaigned against the Australian activist's extradition.
Assange was denied permission this month to appeal at Britain's Supreme Court against a decision to extradite him to the United States. He could still challenge any decision from the government to approve his extradition.