Certified GKH Gardening Expert
Depending on whether the roots were damaged, the plant may still recover. First, move the cactus indoors and make sure it doesn't get any more cold damage. Check for any black or brown portions that may develop and promptly cut those off at the nearest joint. Make sure not to overwater, since it may use less water in its damaged state. Also see this article:
You could also try taking cuttings to regrow the plant in case it doesn't survive.