We purchased a 4 1/2' x 2' piece of granite (3/4" thick) to use as a bar. We would like to hang it from the wall without a base and are trying to find the best possible solution.

The quick and dirty way is to use steel shelf brackets, making sure the brackets are anchored to studs, and glue the granite to the brackets.

The really slick way is to have steel brackets made to be installed inside the wall, bolted to studs, and extending out through the wallboard under the granite. These will take some careful detailing, and will require some mild destruction to the wall to install. A good decorative iron shop should be able to design and make the brackets for you. Two of them will be sufficient.