Immigration Law Questions? Ask an Immigration Lawyer.
Assuming they have met all of the requirements for common law marriage in Alabama, they can be recognized as such under the laws of that state. For immigration purposes, however, they would need to obtain an actual marriage license.
There is no prohibition in federal immigration laws to illegals marrying citizens, but some jurisdictions will require proof of immigration status. Others, however, may only require a picture identification.
Once they are married, the U.S. citizen can file an immediate relative petition for the illegal, but the application could not proceed to adjustment of status (green card) because of the illegal entry. In order to obtain residency, the illegal would need to wait for the old amnesty law to be extended (last extended to April 30, 2001) or for a new one to be enacted.
Best of luck!