On the Batch Process page there are option tick boxes for Same as Source File Location and Overwrite Existing Files. They should do the job if you don't add any Prefix or Postfix to the file names.
Thanks Ryan for the info. I see that check off feature but that's not what I was looking for. In Adobe Audition 1.5, they had a Delete Spot box to check off under the Batch Processing Tabs. At my company, we move raw audio from one folder to a cut/checked folder, and the 1.5 system would delete the "old" spot so i wouldn't have to stop, go back and chlean up that folder. Unfortunately, the CS 5.5 series, doesn't seem to have that feature built into it.