Find a Mediator
Some workplace clashes turn into intractable battles worthy of a war movie. If you find yourself locked into a fight with one person, it may be time for the two of you to bring in a mediator.
You can introduce the conflict to a mediator formally or informally. An informal meeting with an objective person who both parties respect can help you talk through problems without the formalities (and potential subsequent consequences) of a formal mediation.
If an informal meeting doesn't put the issue to rest, you may need to contact a supervisor for help in resolving the conflict. If you choose a formal mediation, be sure to be on your best professional behavior. Don't attack your opponent. Present the conflict in objective terms and not as an emotional reaction to someone who happens to have an opposing viewpoint.
In a formal mediation, expect enlightenment. A third-party observer may be quick to point out failings in your argument that you just could not accept from a co-worker you dislike. And that's the beauty of mediation -- it helps bring the conflict to an objective, level playing field that's fairer for everyone.