Is it possible for someone who is not a green card holder to become a US citizen by serving in the US Army?

IF a foreign citizen wants to volunteer to serve in the U.S. armed forces, they can, (if they are resident alien of the U.S.)BUT they have to apply from with-in the U.S. so they still need a Green Card to enter the U.S.A. to apply to serve.
For your information, about 15 percent of the U.S. troops in Iraq are NOT U.S. citizens, many are from South and Central America, and they hope to get U.S. citizenship by serving a 4 year enlistment. Kind of sounds like the Roman Legions, who were NOT Romans at all, but fought for pay.
*attention:* NO. You need to be at least a legal resident in order to apply to any of the military corps. Go to a recruitment office an talk to them. You will see for yourself. If you lie about your status or social security number in the application, you will be punished by the law, and will be charged with a felony, and then never and ever can you apply for any immigration benefits at all.

CLARIFICATION: in order to enlist in the US military, you must have a green card. Within one year of your enlistment you will be able to apply for citizenship, as opposed to 3-5 years for civilians. The processing fee is waived for active duty military, and expedited as you will already have had a stringent background check. And, you only have to file the N400 and one other form and the military will do the rest. Detailed instructions are on the USCIS website.