How to move from 32bit SQL Server install to 64bit?

A production server has accidentally been installed with SQL Server 2012 32bit version, while it needed a 64bit version.

How best to fix this issue?

Is it possible to simply UPGRADE from 32 to 64 bit version?

Should I just install a new instance of 64 bit version of SQL Server 2012 and then simply restore the backups?

As stated in BOL here:

Cross-platform upgrade is not supported. You cannot upgrade a 32-bit instance of SQL Server to native 64-bit using SQL Server Setup. However, you can back up or detach databases from a 32-bit instance of SQL Server, and then restore or attach them to a new instance of SQL Server (64-bit) if the databases are not published in replication. You must re-create any logins and other user objects in master, msdb, and model system databases.

If you have installed a default instance of SQL Server and wish to keep it that way your only option is to remove the current install and install using the 64-bit media. Otherwise you can install a 64-bit instance as a named instance on the same server.

Cindy Gross has some additional useful information here.

StackExchange Database Administrators Q#55030, answer score: 8