In string theories, extra dimensions must be compactified. The possibility that gravity can have large radii of compactification leads to a violation of the inverse square law at submillimeter distances. We are preparing a null test of Newton\'s law with a resolution of one part in one thousand at 100 micrometers, which will probe the extra dimensions down to a few micrometers. The experiment will be cooled to 4.2 K. To minimize Newtonian errors, a near null source in the form of a circular disk of large diameter-to-thickness ratio is employed. Two test masses, also disk-shaped, are suspended on the two sides of the source mass at a nominal distance of 100 m. The signal is detected by a superconducting differential accelerometer. We are also designing a space experiment which will improve the resolution by more than two orders of magnitude over the ground experiment.