How severe an allergy is it?
If you have ever had difficulty breathing due to your throat closing up (anaphylaxis), please have your doc prescribe you an Epi Pen. This is an injection pen that you use if you are having trouble breathing through your throat and lungs. It provides a short acting treatment to buy you time to get to an emergency room. It only lasts long enough to get to help, not long enough to get you through an entire anaphylactic allergy attack.
Any allergy desensitization should be done with a qualified allergist; the home remedy of living with a cat is ill-advised if you don't have the Epi pen to help you get to a hospital.
If the cat will allow it, it is possible to bathe it and remove a large amount of allergen (cat spit, skin cells, etc).
Using a HEPA air filtration system may further reduce the airborn particles which may trigger allergy. Diligent vacuuming (also with HEPA filtration) helps. Reducing soft fuzzy coverings that hold dust helps - carpet is basically a dirt sponge in disguise, drapes hold a fair bit of dust too. Use bedding which may be washed in hot water (weekly if not more often). Keep any cat/pets out of the bedroom. Use dust protectors on pillows (reduces dust mites, which just build on any other allergy).
It can be pretty rigorous to keep the home environment in a reduced allergen state.