If you completely exit the 1Weather application and return, do the ads still appear? The in-app purchase should remove the ads, but sometimes there is a slight delay between purchase and when this happens. If you still see ads after returning, you may want to try rebooting your device. That will usually remove them quickly.
Sometimes ads will return briefly after an update is you launch the app before digital rights management has time to run. Usually if you exit and re-enter, or reboot your phone, the ads will clear.