Magnetars are neutron stars with ultrastrong magnetic fields, which can be observed in x-rays. Polarization measurements could provide information on their magnetic fields and surface properties. We observed polarized x-rays from the magnetar 4U 0142+61 using the Imaging X-ray Polarimetry Explorer and found a linear polarization degree of 13.5 ± 0.8% averaged over the 2? to 8?kilo?electron volt band. The polarization changes with energy: The degree is 15.0 ± 1.0% at 2 to 4 kilo?electron volts, drops below the instrumental sensitivity ~4 to 5 kilo?electron volts, and rises to 35.2 ± 7.1% at 5.5 to 8 kilo?electron volts. The polarization angle also changes by 90° at ~4 to 5 kilo?electron volts. These results are consistent with a model in which thermal radiation from the magnetar surface is reprocessed by scattering off charged particles in the magnetosphere. Magnetars are young neutron stars with high magnetic fields that are usually observed at x-ray wavelengths. The emission mechanism and geometry of the emitting region have been unclear. Taverna et al. measured the x-ray polarization of the magnetar 4U 0142+61. The polarization degree and angle change as a function of x-ray energy, indicating two different emission regions. The authors preferred a model in which most of the x-rays are emitted by an equatorial band on the surface of the neutron star, with some of the photons then being scattered to higher energies by collisions with electrons in the surrounding magnetic field. ?KTS Measurements of a magnetar?s x-ray polarization constrain models of the emission mechanism.