In short:

Login to the Verba webapp and stop verba services so they don't write into the database.
Repoint the web application's database to the future one so you can login after the database is moved.
Move the Verba database.
Login to the Verba webapp and repoint all server's database and start the services.

Step 1 - Login to the Verba Web Application and navigate to the Administration > Verba Servers.

Step 2 - Go through all of your Verba servers you can see on the Find and List Verba Servers page and do the following steps for each:

Click on your server on the list to enter it's Verba Server Configuration page.
Select the Service Control tab and make a note of the currently running services
Select the Service Activation tab and make a note of the currently enabled services.
Disable all services by clicking on the gearbox with a red cross on it on the rigth side of the page.

Step 3 - After all services are stopped (no services write to the database ) select your Media Repository Server from the Find and List Verba Servers page.

Step 4 - Go to the Change Configuration Settings tab and on the page find the following node: Database Connection

Step 5 - Modify your Database Hostname to reflect your future hostname and in case you are moving your database to an SQL cluster also select your Database Driver (ODBC Driver 10 or 11).

Step 6 - Click on the floppy icon on the top middle of the view to save your changes then follow the instruction on the yellow stripe to apply the changes to the server.

Step 7 - Move your SQL database to the new location (using SQL Server Management Studio):

In SQL Server Management Studio Object Explorer, connect to the instance of the SQL Server Database Engine from where you want to move your database and then expand the instance.
Expand Databases, and select the name of the user database you want to detach.
Right-click the database name, point to Tasks, select Take Offline then click Detach. The Detach Database dialog box appears.

The newly detached database will remain visible in the Databases node of Object Explorer until the view is refreshed. You can refresh the view at any time: Click in the Object Explorer pane, and from the menu bar select View and then Refresh.
In SQL Server Management Studio Object Explorer, connect to the instance of the SQL Server Database Engine to where you want to move your database and then expand that instance.
Right-click Databases and click Attach.
In the Attach Databases dialog box, to specify the database to be attached, click Add; and in the Locate Database Files dialog box, select the disk drive where the database resides and expand the directory tree to find and select the .mdf file of the database; for example: C:\Program Files\Microsoft SQL Server\MSSQL13.MSSQLSERVER\MSSQL\DATA\Verba.mdf

Alternatively you could use the backup-restore steps in SQL Management Studio to move your database.

Step 8 - Login to the Verba Web Application and navigate to the Administration > Verba Servers.

Step 9 - Go through all of your Verba servers you can see on the Find and List Verba Servers page and do the following steps for each:

Click on your server on the list to enter it's Verba Server Configuration page.
Select the Change Configuration Settings tab and on the page find the following node: Database Connection
Modify your Database Hostname to reflect your new hostname
Click on the floppy icon on the top middle of the view to save your changes.
Select the Service Activation tab and activate your previously disabled services.
Select the Service Control tab and start your previously running services.

Step 10 - Follow the instruction on the yellow stripe to apply the changes to all servers. On the Configuration tasks to be executed page check the Check All checkbox to apply the changes to all of your servers.

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