Veeam Backup & Replication Console Fails to Open After a Hostname Change

After changing the hostname of the Veeam Backup Server, the Veeam Backup & Replication Console fails to connect, displaying the error:

No connection could be made because the target machine actively
refused it 127.0.0.1. Make sure the backup server you are connecting with
is X.Y version.

_{The error message displayed by the Veeam Backup & Replication Console mentions a potential version incompatibility because the console assumes that it cannot contact port 9420 because the wrong version is installed. In fact, the reason it cannot reach port 9420 is that the Veeam Backup Service service has failed to start.}

When the Veeam Backup & Replication Console fails to launch in this manner, you will find that the Veeam Backup Service is stopped and cannot be restarted. The diagnostic logging on the Veeam Backup Server will contain the following messages, which show that the service is attempting to connect to the SQL Server Instance using the previous hostname.

%programdata%\Veeam\Backup\Svc.VeeamBackup.log:

 Info (3)    [MSSQL] Connecting to SQL Server (CurrentUser=[NT AUTHORITY\SYSTEM], ServerInstance=[oldhostname\VEEAMSQL2016], Options=[Master], Environment=[Default])
Error (3)    [MSSQL] Failed to connect to server oldhostname\VEEAMSQL2016. (Microsoft.SqlServer.Management.Common.ConnectionFailureException)
Error (3)    [MSSQL] A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: SQL Network Interfaces, error: 26 - Error Locating Server/Instance Specified) (System.Data.SqlClient.SqlException)

The Veeam Backup Service is failing to start and, consequently, preventing the Veeam Backup & Replication Console from connecting because the configuration database connection settings are set to connect to the SQL Server instance using the old hostname, which has been changed.