Two password prompts on VH alias?

Here's the setup:

Apache 1.3
Virtual host example.com, with alias www.example.com
Directory /protected which requires basic authentication

Here's the behavior:

If you type in http://www.example.com/protected/, you're given a password
entry window and then *another one* after which you find yourself at
http://example.com/protected/.

Is this how it's supposed to work?

Thanks,
-Dan