Yes. But while it's possible it's really difficult. You see, it's based on two tiers.
Tier 1 is high school graduates or people with "equivalent".
Tier 2 is GED, some home schools, some online high schools, etc
Tier 2 spots are RARE but they do come up. BUT if you get 15 college credits to go with your GED you'll be considered tier 1 as having an "equivalent" education.
(Generic GED statement)
The US Army breaks down education qualifications into Tiers for enlistment. Tier 1 includes high school graduates while Tier 2 includes GED credentials.
Tier 1 applicants can apply for any open slots and require a 31-percentile* or better on the AFQT**.
Tier 2 applicants may only apply for a much smaller number of slots which are only opened at the needs of the Army. Additionally, GED holders must score above the 50-percentile on the AFQT. These slots usually close quickly--when available--because of the number of GED holders applying.
However, if a GED holder earns 15 semester hours of college courses (approximately 1 semester, full time), the US Army redefines them as Tier 1 for enlistment purposes and allows them to apply for any open slots without the restrictions of the Tier 2 applicants. The college courses used must be from an accredited school***, must be applicable (though not necessarily applied to) a degree, and must be at the 100-level (Freshman-level) or above.****
(The above information is from AR 601-210 Active and Reserve Components Enlistment Program in Section 2-7 starting on page 8.)
Usually, the best source for these credits will be your local community college. Most community colleges have class schedules that allow people to work while completing courses as well as a lower price in-line with typical student aid packages. Community colleges also tend to have courses available (and other academic support) to help people with difficulties in basic skills like math get up to speed.
You can start by completing the Free Application for Federal Student Aid (FAFSA) on-line (http://www.fafsa.ed.gov/) and contacting the admissions office at the local community college. FAFSA forms the basis for being awarded student aid, such as grants (you don't have to pay back), scholarships, and student loans.
* - Percentiles (abbreviated %ile) are a measure--unlike percentages--which compares how test-takers rank. In other words, if someone scores a 50-%ile on a test, they scored as well as exactly 50% of the test-takers regardless of actual score.
** - AFQT or Armed Forces Qualification Test is a subset of the ASVAB where scores are calculated as a percentile and used to determine trainability of recruits. A generic explanation is here: http://mldc.whs.mil/download/documents/Readings/ASVAB_4.pdf Other information can be found here: https://www.march2success.com/index.cfm
*** - School accreditation is searchable through the Department of Education (http://www.ope.ed.gov/accreditation/). The recruiting office uses--as sources--the Accredited Institutions of Postsecondary Education (AIPE) book published by the American Council on Education and the National Association of Credential Evaluation Services (NACES) which generally parallels the Department of Educations listings. Consult a recruiter if there is a question about accreditation.
**** - An additional benefit of these college courses is the fact they count towards civilian education points toward promotion for E4's and E5's. Also--depending on your course selection--they may improve your ASVAB performance or complete academic pre-requisites for some MOS's. Consult with a recruiter about what pre-requisite courses apply to particular MOS's.
http://www.usarec.army.mil/hq/apa/download/USAREC_TPs7-12.pdf As of July 1st, 2012 you can join with a GED as Tier 1