Wadi Rum and the Dead Sea both encompass many miles and are 188 miles (303 km) apart. Travel by car with a private driver is your best option. For planning purposes, two useful reference points at each location are Wadi Rum Village and Amman Beach—the most direct route between them takes about four hours. You can also hire a taxi instead of arranging private transport in advance, and it's possible to hire the driver for the day to give you more time to explore.
Buses are available but the process will be quite time-consuming. You'll first have to catch a ride (minibus or taxi) to a larger town like Wadi Musa (Petra), Aqaba or Amman. From these places, you'll have more options for tourist buses, minibusses or shuttles to continue on your journey.
By Private Transfer
Duration: 4-5 hours, with no stops
Having your own driver means freedom and flexibility for your journey onto the Dead Sea. The route takes you north via the modern Desert Highway and Route 65 which hugs along the coast. Once you leave the Wadi Rum area, you'll travel a mix of quiet, scenic roads and modern highways. If you can, consider detouring to take advantage of the wilderness areas and notable sites along the way—Petra, the Dana Biosphere Reserve, and the Crusader-era Shobak Castle are all options.
Once you reach the coast, you'll see there are a variety of ways to experience the Dead Sea. You could opt for a private beach at a luxurious spa resort or visit the popular (and public) Amman Beach. If you're interested in "off the beaten track," your driver will likely have helpful suggestions about more secluded places to visit.
Duration: 4-5 hours
The village of Wadi Rum is fairly small so it may be best to book your taxi ahead of time if possible. Your park camp can help do it for you before you depart. There may be a wait, too, if the driver is coming from another town. Most drivers will want a full vehicle so you may have to wait for other travelers before you can leave.
Sharing a taxi can be a cheaper rate but since this is a long ride, consider negotiating your own driver for a half or full-day. It can make for a less-crowded vehicle and a good way to get local insight into the places you'll be passing on your journey.
Duration: 6-8 hours
Getting to the Dead Sea by bus will take a good portion of a day so it's not ideal if you're on limited time. Again, your camp guide can help coordinate this transportation if need be. Since no buses run directly to the coast, it's a good idea to figure out where you'd like to go and then decide which route would get you there quickest. The buses out of the village run daily but infrequently.
If you're traveling on a Friday, inquire about service as most public transport doesn't run that day. Once you're in a bigger town, you'll have more available to you. Consider upgrading to a comfortable JETT tourist bus for the long ride since they have air-conditioning and comfy seats.
If you're heading to a particular hotel resort, call ahead and see if they offer a shuttle service from other locations.