If the pipe is still frozen at this time AND is made of metal (copper or galvanized) I would recommend that you continue trying to thaw the hot water pipe using the hair dryer. Here, metal pipe can act as a conductor of heat and may conduct the heat far enough along the length of the pipe and into the wall to thaw that pipe. Keep the dryer about 1" off the pipe and it may take 15-20 minutes (so give the hair dryer a break every 5 minutes or so), but in a lot of cases this does the trick.
The single most important thing to know when thawing a pipe is to be sure that the hot tap is fully opened. In this manner, as the pipe starts to thaw you will see some dripping start to occur and that means that you are close and to just hang in tight... pipe will thaw quickly from there...
As Hkstroud mentioned, if the pipe thaws and is burst it will be obvious as you will hear a firehose running behind the wall. In this case, you'll need to know where the main water shutoff is (or the cold water valve into the water heater) so you can shut the water off and call a plumber to make the repair!
In the future, keep the cabinet open on those extremely cold nights or, if absolutely necessary, leave the hot tap with a slight drip overnight... should keep this from happening again!