The Republic of Ireland international has featured 33 times for the Baggies since arriving on loan from Brighton at the start of the season.
The Express & Star revealed last month Molumby had triggered an appearance-based clause to makes that loan move permanent.
And the club have now confirmed that is the case.
The fee for Molumby’s transfer is officially undisclosed.
But the Express & Star understands it is in the region of £900,000.
The Baggies were keen to keep Molumby because they believe he is a player with the potential to improve significantly over the next few years.
Despite being just 22 years old, Molumby has already won 12 caps for the Republic of Ireland.
“We’re really pleased to have Jayson on board permanently,” manager Steve Bruce said.
“He’s a young player full of energy and passion, and we see plenty of potential in him to grow and develop further.
“He’s only 22 and he’s already played more than 75 games in the Championship.
“He’s managed to force himself into our XI recently and now his challenge is to stay there by continuing to show improvements in his game.”