We describe a nuclear spin gyroscope based on an alkali-metal-noble-gas comagnetometer. Optically pumped alkali-metal vapor is used to polarize the noble-gas atoms and detect their gyroscopic precession. Spin precession due to magnetic fields as well as their gradients and transients can be cancelled in this arrangement. The sensitivity is enhanced by using a high-density alkali-metal vapor in a spin-exchange relaxation free regime. With a K-3He comagnetometer we demonstrate rotation sensitivity of 5 x 10(-7) rad s(-1) Hz(-1/2), equivalent to a magnetic field sensitivity of 2.5 fT/Hz(1/2). The rotation signal can be increased by a factor of 10 using 21Ne with a smaller magnetic moment. The comagnetometer is also a promising tool in searches for anomalous spin couplings beyond the standard model.