The versatile left-sided player is able to move outside of the transfer window as he is a free agent after being released by Sunderland at the end of last season.
Dossena, a former Italy international, has also had spells at Udinese, Liverpool and Napoli and he joins an Orient side 20th in League One.
"Orient have agreed a contract with former Liverpool, Napoli and Sunderland left-sided player Andrea Dossena," read a statement on the club's official website.
Dossena's first taste of English football came when he moved to Anfield for a reported £7 million fee in July 2008.
Dossena's stay on Merseyside proved brief, though, and he joined Napoli in January 2010.
A loan spell at Palermo followed before he spent the 2013-14 campaign at the Stadium of Light.
