Certain exact properties of the stationary force-free magnetosphere of an axisymmetric pulsar are obtained. In particular, it is shown that a magnetic separatrix has an inclination angle of 77.3 degrees to the equatorial plane. The electromagnetic field has an R(-1/2) singularity inside the separatrix near the light cylinder. A numerical simulation of the magnetosphere which crudely reproduces these properties is presented. The numerical results are used to estimate the power of an axisymmetric pulsar: L=(1+/-0.1)mu(2)Omega(4)/c(3). Thus, the background magnetic configuration, on which all the puzzling pulsar phenomena are taking place, is now known with some confidence.