If you lose your PRO license immediately after purchase, the purchase is not recognized.
It can sometimes take a few minutes after purchase, reinstall, or app update for the Play store DRM to run and for Play to recognize that the app has been purchased. If you are still seeing ads after a time, the first thing to do is reboot your device.
If that doesn't make a difference, go into the application manager/manage applications in your device settings and locate the Play app and then clear the cache for Play.
This should force the Play app to re-query its database for purchases.
If you are seeing ads after the update please select the Restore to AdFree option in the 1Weather menu and click restore.
If still, the ad-free version is not restored open 1Weather -> Menu -> Remove ads. Now press $1.99 in the dialogue box that appears. If you have only a single google account on your device, this step will typically make google play recognize your purchase, and the Ad-free version gets restored.
If you have multiple Google accounts, and the linked google account is different from the one you had used to purchase the app, this takes you to the next screen. On the next screen, check which mail address is shown. Note that you need not buy the app; check the email address shown.