Can I use mirroring to make a copy of my Database? We are moving a database off of a older physical server and putting it onto a Virtual Machine (SQL 2005 to 2014 I hope).  I need to copy a single database from the physical to virtual with as little downtime as possible. Originally I was just going to do a full backup and then restore on the new machine, but that will take almost an hour on this database.

My question is: If I can get the new virtual machine setup before the downtime, would it be effective to setup a database mirror from the old Physical box to the new Virtual box?  Then when the downtime came I would cut the mirror (database should be in sync) and realias the server to be the same name as the old physical database server?

I have never had to use mirroring so I was not sure if this was a "Smart" way to do it. If it would work the only downtime would be the name change.
The main problem with mirroring is that if you have a network outage or get disconnected mirroring will stop. So across networks it is better to use log shipping. 

Additionally it sounds like you have a vldb or poor disk i.o based on your time listed in your question which is why you might prefer log shipping for the migration as you would only need to restore the last log to bring up the new database. This is ideal for minimizing downtime.

This blog, while old, highlights some points i explained