Using first-principles theory, we have calculated the energy of Gd as a function of spin direction, theta, between the c and a axes and found good agreement with experiment for both the total magnetic anisotropy energy and its angular dependence. The calculated low temperature direction of the magnetic moment lies at an angle of 20 degrees to the c axis. The calculated magnetic anisotropy energy of Gd metal is due to a unique mechanism involving a contribution of 7.5 microeV from the classical dipole-dipole interaction between spins plus a contribution of 16 microeV due to the spin-orbit interaction of the conduction electrons. The 4f spin polarizes the conduction electrons via exchange interaction, which transfers the magnetic anisotropy of the conduction electrons to the 4f spin.