Weather without location services, using just one fixed location
First of all, open weather.app and tap the bulleted list in the bottom-right.



Then click the plus button and search for your location. This will then appear at the bottom of your list. Hold your finger on it and then slide it to the top of the page (underneath your current location).

Then go to Settings -> Privacy -> Location Services -> Weather and click Never.



Then when you return to the weather.app the location should appear at the top and be the default.