Windows Update was failing to install several updates Office 2003( KB2502786, KB2509503, KB2522981, and KB2464588). I eventually found the answer. I opted to manually update the registry. I only had to apply the first registry change described in KB2258121 for the updates to work.
This article (KB2258121) provides a fix for a problem in which you receive error code 0x80070646, error code 646, or error code 1606. This update issue occurs when you install Office updates by using Windows Update or Microsoft Update. For detailed information, see the “Symptoms,” “Cause,” and “Resolution” sections of this article.