That seriously limits your options if you can't setup a crontab. I'd say the short answer is no, you can't setup the code to run once a week.
The long answer is, at least not serverside. You could setup another box with a crontab or AT (windows) command, which launches a browser that visits a page that executes your code once a week... then just add in some javascript in your code that closes the window when the code is complete (or you could probably make cron or windows just close the browser after x minutes too)
-Rob
(maybe I'm wrong... but that's my understanding)