This is a difficult issue. Here are my suggestions.
1. The bid is valid for the one year period between September 26, 2008 and September 25, 2009.
The refers to the specific period of time mentioned (one year).
2. The bid is valid for a one year period between September 26, 2008 and September 25, 2009.
A refers to a period of time (again one year): however it doesn't refer to the specific period mentioned, but instead indicates 1 year as a general unit of time. It can be thought of as a generic referrence.