Congratulations. Marriage is a beautiful thing!
Don't know what your budget is, as everybody's idea of "cheap" is a little different, but I can give you my experiences.
My wife and I spent our honeymoon at Sandals on Grand Bahama. This is a GREAT honeymoon spot, but not that cheap. With the specials it will probably put you at about $3500-4500 for a week for both of you, but it does include diving at an adults only resort with all food and drinks included. If you drink a lot, it's easy to justify this price on the bar tab and other inclusions.
At the other end of the tropical Caribbean spectrum is Cozumel. My wife and I spent around $2400 there last November including diving, food and drinks and airfare. This was at the semi-upscale Iberostar resort. You can go way cheaper if you're willing to stay in one of the smaller places that maybe aren't beach front. I've seen package deals that are as low as $619 per person at the Casa del Mar (
Caradonna Dive Adventures - Dive Travel and Vacations) including food and diving, no airfare. You can also check packages at Blue XTSea Diving. Christi, who runs that outfit (a regular member here), usually has good package deals. I just checked and she must be updating because prices aren't posted, but here's the link: (
Cozumel Diving: Blue XT SEA Diving, Cozumel Mexico).
Anyway, congratulations again, and best of luck!