O's boss Paul Brush was determined to strengthen his squad at Brisbane Road and has moved for Alexander from Division Three rivals Hull.
The Tigers had said that they would be willing to let him leave the club following Peterborough's attempt to sign him on loan recently.
Orient have now come up with a satisfactory offer for the 23-year-old and Tigers chairman Adam Pearson was happy to allow him to leave the Kingston Communication Stadium.
"I would like to wish him all the best in his future career," Pearson said of Alexander.
"Gary is from the London area and I'm sure a move back down there will be beneficial to him."
The former Swindon Town and Exeter man will go straight into Brush's squad for their Division Three clash with Cambridge on Tuesday.
