Is it bad for your health to drink one diet soda a week?

Probably not: But its better to skip it.

For teeth... yes: Diet soda contains corrosive acids such as citric, malic, tartaric carbonic or phosphoric which, can remove calcium from tooth enamel. Without protection from the dental enamel, the tissue underneath is susceptible to cavities, root canals & even extractions. Sugar is an additional cavity causing issue, so consider rinsing your mouth after drinking soda, both diet & soda containing sugar.

It's OK: If your a healthy person with good oral hygiene habits, i think your body and teeth can handle one soda a week. Enjoy that soda!