Connection management is done deep within the Database class so it's hard to modify the behavior without creating a new Database implementation and overriding the CreateConnection method (you may also need a custom DbProvider as well). This
answer shows how.
Another option would be to work with the connection directly using ADO but it seems like you want to test using the Enterprise Library code.
Your best bet might be to enable connection pooling and, since this appears to be for a test, set the min and max pool size to 1 which would force the same connection to be used and also set Connection Reset to false.
I'm just curious what you are trying to accomplish with only having one connection?
Enterprise Library support engineer