If your database structure allows for it you could look at peer to peer replication(PTP). You can use it to essentially build a two node active/active cluster. Once PTP replication is in place you will have two live copies of the database both able to accept new data while updating the other. This will allow you to switch over your applications with out having to take the database offline.
The tricky part of this is going to be getting your two databases in sync long enough to build up the PTP replication. There are also some caveats around your primary keys, specifically for any new rows. This will require that each table generates a unique key whether it be guid's, identity increments by 2 or more or building unique keys specific to the application. This in theory could allow for a no down time cut over but it will be tricky.
As others have posted, mirroring is a relatively quick and easy way to cut over. But it will require a few seconds of down time at least - this down time can be kept to a minimum if everything is scripted out appropriately.