The glamour model, who divorced singer Peter Andre last year, tied the knot with Celebrity Big Brother winner Reid at the Wynn Hotel wedding chapel in front of family and friends.
The newlyweds said in a statement: "We are very much in love and look forward to the future together. We can't wait to get back and celebrate our marriage with our friends and family who we know fully support our wishes."
A guest told The Mirror: "Kate was incredibly excited. She had the idea while Alex was in the Celebrity Big Brother house and just decided to run with it. She loves Alex and he is great with her three kids, so she thought she'd go for it.
"They may have secured a lucrative [magazine] deal for the rights to their wedding, but their love is genuine. Alex worships the ground she walks on and Kate fancies the pants off him."
After being pronounced man and wife, it is thought that the pair were snapped by a magazine photographer before returning to their penthouse suite at the five-star Planet Hollywood Resort & Casino.
Price and Reid were first romantically linked in July last year.