In the US, one gains trademark rights through use of the mark in commerce, not through registration. If you began using the mark before the other company, you would have superior rights. One exception is if the other company filed their trademark application before you began using the mark. Please note that just owning a domain name is not the same as using the mark in commerce.
You really do need to consult an attorney on this one; it is best not to reply to the other party on your own.
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If certain conditions are met, yes. I suggest that you speak with a local business lawyer before you take any action. You can use Avvo's "find a lawyer" tool to assist you.
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I agree with the other answers. I will add the following: there is a difference between registered trademarks and a common law mark that arises from use in commerce. You may need an attorney to assist you in this dispute, but you can also get some quick answers to your questions here: http://www.uspto.gov/faq/trademarks.jsp#_Toc275426680. Also, as another attorney mentioned, use of a trademarked name in a domain name is not enough--you have to use the mark in commerce. However, I will add that if you also have a domain name dispute on your respective websites, you can retain the services of an attorney in a WIPO arbitration proceeding. This is a relatively efficient and inexpensive process. see http://www.wipo.int/amc/en/domains/
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