Your best bet is to take a taxi. You should be able to negotiate the trip, one-way, for about RD$1000-1200 and 1.5-2 hours (RD$15.8-16 = US$1). Many of the drivers that work that airport live in Higuey, which is half-way to Punta Cana, so you might be able to get a better price by using one that plans on going home to Higuey after the drive, since it won't be so far for him to come back.
Renting a car is an option, but I would not recommend making that drive at night.
Search the archives, as all the alternatives for this route have been discussed.
Jim Hinsch JimHinsch@CSI.COM
