Hi!
Check out this thread: http://bestdestinationwedding.com/forum/t933 . Here you will see how much people spent on their entire wedding.
Your question is difficult as everyone has a very different definition of 'cheap.' Like you, my dream wedding is the cheap one where I still have money left over to start a comfortable life. My fiance and I are also footing the entire bill.
Your best bet might be to find a resort with a free package and add on stuff as needed -- that consistently seemed to be the cheapest way to go. The RIU resorts have free packages and have semi-private receptions included with them that may suit your needs. There are a few RIUs in the DR and a couple of the resorts are relatively inexpensive (I am thinking of RIU Tiano and RIU Nabioa). The only catch is 80% of your wedding party must stay there and you must have a minimum amount of rooms booked before the wedding is free AND there is a per-person charge after 10 guests. Dreams resorts also have free packages and the cost seems to be directly proportional to the number of guests and not as reliant on people being there. There are extra charges for private receptions and semiprivate receptions, while free, are not guaranteed. Oh, and almost anywhere in DR, you will have to pay $300 for the legal fees regardless of whether the wedding is 'free'.
BUT that is just 2 of MANY resorts to choose and it really depends on what you're comfortable with. You just gotta do the research and price out what you want.