Moving an SQL Database from Windows 2003 - 2008

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I've got a SQL server 2008 database here which currently is used with our encryption software installed on Windows server 2003. Now the server is dying and we need it transferred onto a new 2008 Server but the problem I have is that I've never used SQL and I'm not sure if I've done it right or if there are thing's Ive missed in the process.

 

My thoughts were to create a backup of the database and then restore it on the new server. Currently (2003 server)when connecting to the database it connects showing ServerName\DatabaseName. However, if I try to create the database on the new server it wont let me create an instance with ServerName\DatabaseName but it will let me create DatabaseName which I can then restore the database to. When I try to install our encryption software and point it to the new database it provides me with an error saying that the database is either not there or not configured for remote access. I assume this is because I have not restored the database using the same ServerName\DatabaseName which means I haven't configured SQL correctly.

 

Unfortunately this was originally setup by someone who is now off and as the server is dying it needs to be moved ASAP.

 

Is there something I can check to allow me to create a new database with the same structure as SererName\DatabaseName?

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OK, it looks like I haven't setup SQL server correctly so back to the drawing board it is.