Alkali-metal magnetometers compete with SQUID detectors as the most sensitive magnetic field sensors. Their sensitivity is limited by relaxation due to spin-exchange collisions. We demonstrate a K magnetometer in which spin-exchange relaxation is completely eliminated by operating at high K density and low magnetic field. Direct measurements of the signal-to-noise ratio give a magnetometer sensitivity of 10 fT Hz(-1/2), limited by magnetic noise produced by Johnson currents in the magnetic shields. We extend a previous theoretical analysis of spin exchange in low magnetic fields to arbitrary spin polarizations and estimate the shot-noise limit of the magnetometer to be 2x10(-18) T Hz(-1/2).