A great deal of evidence has recently been gathered in favor of the picture that soft gamma repeaters and anomalous X-ray pulsars are powered by ultrastrong magnetic fields (B > 10(14) G; i.e., magnetars). Nevertheless, present determination of the magnetic field in such magnetar candidates has been indirect and model-dependent. A key prediction concerning magnetars is the detection of ion-cyclotron resonance features, which would offer a decisive diagnostic of the field strength. Here we present the detection of a 5 keV absorption feature in a variety of bursts from the soft gamma repeater SGR 1806-20, confirming our initial discovery (Ibrahim et al.) and establishing the presence of the feature in the source's burst spectra. The line feature is well explained as proton-cyclotron resonance in an ultrastrong magnetic field, offering a direct measurement of SGR 1806-20's magnetic field (B approximate to 10(15) G) and clear evidence of a magnetar. Together with the source's spin-down rate, the feature also provides the first measurement of the gravitational redshift, mass, and radius of a magnetar.