The Baggies made an initial approach for Amalfitano last week, and now the two clubs have agreed terms.
Marseille have confirmed that they will loan the 28-year-old to West Brom for the season with an option to buy him at the end of the season.
West Brom coach Steve Clarke was delighted to see the deal finalised.
Head coach Steve Clarke said: "I'm delighted to welcome Morgan to the club.
"He is predominantly a right-sided midfielder who can also operate in a number of other midfield and attacking positions.
"Morgan is a technical player with good quality on the ball. He also has fantastic work-rate and we believe his game is well suited to the Barclays Premier League.
"Morgan's arrival will increase competition in midfield and add more quality and depth to the squad."
French international Amalfitano joined Marseille in 2011, but he has become surplus to requirements under Ellie Baup and has now accepted a move to The Hawthorns.