The 22-year-old was a free agent after leaving Brighton late last year and has been training with the O’s in recent weeks.
Smith impressed as the Seagulls won promotion from npower League One last season but could not break into Gus Poyet’s side in the Championship.
He worked under O’s manager Russell Slade when the pair were at Brighton and has now put pen to paper at Brisbane Road.
Smith could go straight into the Orient squad for Saturday’s trip to Tranmere.
Rotherham boss Paul Warne admits it will be 'bizarre' coming up against Paul Hurst's Shrewsbury in the League One play-off final.