The 17-year- old midfielder, a part-time plasterer, has already played for Blackburn's U18s and earned a 20-minute run-out in the reserve team victory over Manchester United last Thursday.
Marrow was handed his chance after winning a 'keepy-uppy' competition last month.
Chairman John Williams admits it is prudent to explore every opportunity to uncover a potential hidden gem.
"From our point of view, we cannot afford to miss any opportunity to pick up good young players," said the Rovers chairman. "Matt Derbyshire is a good example. It's important that we don't turn our backs on promising young players like him. We must explore every avenue.
"With Alex, he came to us after he won a keepy-uppy competition and he's been recommended by Glyn Hodges (Rovers reserve team boss).
"They already like him down there. They've seen something in him, and we've given him a contract until the end of the season.
"It gives the lad a few months to show us what he's all about, and he's already had a little taste of things in the reserves against Manchester United."