I've found conflicting answers on other website and was wondering if anyone knows.
The Basilica was built next to a bell tower built in 1414. Although the English Wikipedia article states that Alberti's restoration began in 1462, most of the other sources that I found state 1472. Also, Alberti sent a description and a drawing of the proposed site to Gonzago (the patron) in 1470, after 1462.
The construction began in June 1472, shortly after Alberti's death in April, and was carried on by Luca Fancelli. The restoration went on and off for centuries and was completed in 1732. The Western Facade itself was completed in 1488.