It will stop. You are only charged for the droplets that exist on your account. Anytime a droplet exists for less than a full calendar month you are only charged at it’s hourly rate for the time it is on your account. For a 512MB droplet this is $0.0074/hour.

If you are upgrading you can also choose to resize your existing droplet from the control panel. This will allow you to move to the 1GB droplet size while keeping all your files and configuration in place.