The Republican-controlled Legislature passed the law over the opposition of Democrats.
Backers say it will ensure that welfare benefits are used for their intended purpose and not to subsidize drug use and criminal activity. Democrats said the measure places an unfair burden on the poor.
Gerry Weber of the Southern Center for Human Rights said the organization is prepared to file a lawsuit over the issue, but not until it is put into practice. It takes effect July 1.
A similar law in Florida took effect last July but was blocked in October by a federal judge, who cited constitutional concerns.
At least two dozen states have proposed measures this year that would require drug tests for benefits.