Well, I'm sure all the various methods of cooking prime rib (or rib roast) mentioned above all work well for each of the poster, but I have to say that I've
never had a great or very good prime rib anytime when I've cooked it throughout the cooking period at a constant 325 degrees or higher. If that worked for some, that's fine, but I've found the results to generally be very mediocre, at least for me.
Most of the great prime rib restaurants that I've gone to virtually ALL cook prime rib using
slow cook methods, of which there are many, and which generally results in a more tender piece of cooked meat. Some of these methods were mentioned above. Putting the prime rib in a very hot oven, say 400 to 500 degrees, and then
shutting off the oven either right away or within a short period of time, leaving the roast to slowly cook in a completely closed oven for many hours (do
not open the door), is a great, great way to cook prime rib (always use the type of meat thermometer mentioned above -- the kind that Alton Brown uses or as someone above mentioned " . . . a digital probe thermometer . . . the kind that the probe stays in the meat while cooking and there's a wire leading to the temperature readout gizmo."). Afterwards, some put the oven back on low for a little while just before taking the roast out. Alton Brown's method (minus the clay pot) mentioned by some above whereby the roast is cooked at a low 225 degrees (or even lower) and then blasting it at a high temperature at the end for a few minutes is another super way of making a very good prime rib meal. It's worked for me many times.
Let me tell you about a new way (for me) that I "experimented" with last night for the first time. I had bought a 7.5 lb boneless rib roast from Sam's and cut off a small 3 lb piece to cook for just my wife and myself (vacuum sealed and froze the remainder to a time when our "kids" come over). I warmed up our gas grill using just the one burner on the left side and leaving the other burner on the right side turned off. I put a sea salt, pepper, Essence, and steak seasoning rub all over after letting it get to room temperature, inserted my handy dandy probe wired to the "temperature readout gizmo," put the roast upright on the right side of the grill with the readout gizmo on the grill's side table, closed the cover and put the burner on the left side at the lowest heat setting possible. My grill doesn't have a temperature gauge, but it's not needed because of the thermometer that I used. I cooked the roast to an internal temperature of 145 degrees (which took only about 95 minutes since it was such a small piece), opening the cover only a couple times to turn the roast towards the heated side on the left, took it off the grill and let it rest for about 10 minutes. The outside was nicely browned, the little bit of fat on top was nicely crisp, and the inside after carving was a beautiful pink throughout. Best of all, it was so tender, tasty and soft that it melted in our mouths. With a hot baked potato and a few other sides, we had a very nice meal indeed.