We present a design for a novel electronic refrigerator having a base temperature of about 18 mK when operating from a bath temperature as high as 1.5 K. This all-electronic refrigerator is a factor of 104 smaller and lighter than dilution and adiabatic demagnetization refrigerators, and is compatible with conventional photolithographic fabrication. The refrigerator, based on the unique thermal transport properties of a normal-insulator-superconductor (NIS) tunnel junction, preferentially removes electrons whose energy is higher than the Fermi energy from a normal metal. Electrons with an average energy equal to the Fermi energy are returned to the metal by a superconductor contact. Consequently, high energy thermal excitations are removed from the normal metal, thus cooling the electrons. In our configuration, the junction is deposited on a Si3N4 membrane of submicron thickness that thermally isolates the normal electrode from the bath. As a result, both electrons and phonons in the metal are cooled below the bath temperature. We calculate a cooling power of 2 nW at 100 mK, and a base temperature of 18 mK for a refrigerator area of about 100X100 μm2. Using 105 such refrigerator circuits, the cooling power can be increased to 200 μW.