Both forms are correct. Wether and weather are both correct forms. These two words are homophones, which means that they have the same pronunciation despite different spellings, however, in that case, words only differ by one letter. Wether, as well as weather, can be nouns, but weather sometimes also acts as a verb. The only aspect that solely tells these two words apart is their meaning.

Correct spelling, explanation: (used rarely) a male sheep that has been castrated.

Correct spelling, explanation: the conditions in the atmosphere (noun) or to deal with a problem or a difficult situation (verb).

Definition of wether:
noun: a castrated male sheep;
I think that breeding wethers is a cruel process and we should stop castrating animals for our profit.

Definition of weather:
noun: the air conditions in the particular area such as wind, rain, temperature, cloudiness, etc.;
I knew that weather in the mountains can be tricky, but I didn’t expect such severe weather during our trip.
Verb: to successfully live through a difficult situation or a problem;
Gladly, our company weathered all the setbacks that occurred lately and we can begin the recovery plan.
– to change in color or form during a period of time due to atmospheric conditions;
Look how horribly our beach house weathered in the last few years, it is no longer white!