Put the script in question into your crontab:
0 1 * * 1 [[ "$(/bin/date +\%d)" -le 7 ]] && /path/to/script.sh On every Monday at 0100, it will check to see if the date is less than or equal to the seventh, and if so, run the script.
Bonus to using this is it's very easy to reschedule for Tuesdays without having to edit the script.