We demonstrate a general method to measure the quantum state of an angular momentum of arbitrary magnitude. The (2F+1) x (2F+1) density matrix is completely determined from a set of Stern-Gerlach measurements with (4F+1) different orientations of the quantization axis. We implement the protocol for laser cooled Cesium atoms in the 6S_{1/2}(F=4) hyperfine ground state and apply it to a variety of test states prepared by optical pumping and Larmor precession. A comparison of input and measured states shows typical reconstruction fidelities of about 0.95.
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