I don't believe a person is required by law to renounce their citizenship in his/her birth country to become a citizen in this country. The person seeking citizenship in this country is required to pledge allegiance to the United States but can maintain a dual citizenship if they choose.
I do not support dual citizenship because I do not feel a person can swear complete allegiance to one country and not have an allegiance to another country if that person maintains a citizenship in 'that' country as well. The person is divided and his/her allegiance can not help but be divided as well.