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.
The 33-year-old famously scored two goals in consecutive matches for Liverpool against Real Madrid in the UEFA Champion League and rivals Manchester United in the Premier League in March 2009.
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.
