The winger, released by Rangers this summer, spent the latter half of last season on loan at Dunfermline Athletic, making 10 appearances.
Stevens impressed Arsenal during a trial period, scoring for their under-21s against Monaco II.
Arsenal academy manager Per Mertesacker said [arsenal.com]: "Since Archie has been with us, he has really impressed as a person and player.
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"He has fit into our environment seamlessly and has been really impressive in matches and in training.
"Archie has absolutely earned this next stage in his career with his professional contract with us.
"We welcome Archie to our Academy and are all looking forward to working with him in his continued development."
Reflecting on his time at Dunfermline, Stevens said: "We were in a relegation scrap at the start, so that taught me a lot."