If the Rollback transaction is within the Catch Statement, why doesn't the transaction end?

Here's an example script I used for testing this out just now. The table I'm trying to update exists in a different database, so it fails

USE [MASTER]

BEGIN TRY

BEGIN TRANSACTION

Update DBAEmails Set Email = 'thisted@gmail.com' where ID = 2

COMMIT TRANSACTION

END TRY

BEGIN Catch

print 'Failure to Update Email'

RollBack Transaction

End Catch

I get the error message

Msg 208, Level 16, State 1, Line 4

Invalid object name 'DBAEmails'.

But the Rollback doesn't happen?

What you mention in your PS is true. We need to work out a better solution.