Pastry chef Mosshart proposed 28-year-old Osbourne while the couple were holidaying in the Caribbean, reported E! online.
"Mosshart had proposed whilst they were on holiday in the Caribbean. Her engagement ring is designed around a ring her father, heavy mental singer Ozzy Osbourne, used to wear," a source said.
The pair met in 2011 at supermodel Kate Moss' wedding.
Mosshart's connection to Moss is through her husband Jamie Hince. Mosshart's sister Alison is in the same band, The Kills, as Hince.
Mosshart was based in London when he met Osbourne. He has since moved to LA in order to be with Osbourne, who did not enjoy being in a long distance relationship.