A novel technique for microfabricating alkali atom vapor cells is described in which alkali atoms are evaporated into a micromachined cell cavity through a glass nozzle. A cell of interior volume 1 mm3, containing 87Rb and a buffer gas, was made in this way and integrated into an atomic clock based on coherent population trapping. A fractional frequency instability of 6 x 10(-12) at 1000 s of integration was measured. The long-term drift of the F=1, mF=0-->F=2, mF=0 hyperfine frequency of atoms in these cells is below 5 x 10(-11)/day.