If it takes 1 second for your script to run, you need to run it 1 second early ...
Subtract 1 minute from the scheduled running time ...
Put in a "sleep 58" or "sleep 59" line in the top of your cron script
to have it wait almost a full minute before continuing processing
and you may have to play around with it a bit, but you may be
able to time it such that your script finishes running at the
exact time that you want.