Yes.
U.S. law does not mention dual nationality or require a person to choose one citizenship or another. Also, a person who is automatically granted another citizenship does not risk losing U.S. citizenship.
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html
In order to lose U.S. citizenship, the law requires that the person must apply for the foreign citizenship voluntarily, by free choice, and with the intention to give up U.S. citizenship.
—Was I wrong?
Yes.—
I knew this day would come. I had no idea how bad it feels to be wrong. No wonder people try to avoid it. ;->