Well if you want to move the database you are going to need some downtime to do that.
you could keep the database on the old server and then you could have no downtime but I think it would be best to move it off.
Here is what you should do:
1. backup the database
2. Make sure that you can remotely connect to the MSDE on the old box.
3. install the new BES at that same version currently 4.1.4 on the new hardware. (This is where you will need your nd SRP (Migration kit SRP))
4. This will install a second BES into the same database.
5. Now you can start moving users over from the old server to the new server.
6. Once all of the users are moved over, I would backup the database again.
7. Then I would upgrade the new BES to 4.1.6.
8. Once it has been upgraded and everything is still working for a while, I would backup the database again, and get ready to move it over to the new hardware. You might want to follow this RIM KB: BlackBerry Search Results