This article reports the discovery of a luminous (3.5 x 10(37) ergs s-1 over the 0.2-2 keV band) transient X-ray pulsar in the Small Magellanic Cloud (SMC) with an extremely soft component to its X-ray spectrum. This is the first time that a spectrum of this type has been seen in this class of X-ray source. The pulse period is 2.7632 s, and the pulse modulation appears to vary with energy from nearly unpulsed in the low-energy band of the ROSAT PSPC (0.07-0.4 keV) to about 50% in the high-energy band (1.0-2.4 keV). The object, RX J0059.2-7138, also shows flickering variability in its X-ray emission on timescales of 50-100 s. The pulse-phase-averaged PSPC X-ray spectrum can be well described by a two-component source model seen through an absorbing column density of approximately 10(21) atoms cm-2. One spectral component is a power law with photon index 2.4. The other component is significantly softer and can be described by either a steeply falling power law or a blackbody with a temperature kT(BB) approximately 35 eV. This component is transient, but evidently unpulsed, and, for the blackbody model fits, requires a large bolometric luminosity: near, or even several times greater than, the Eddington luminosity for a 1.4 M. object. When these characteristics of its soft emission are considered, RX J0059.2-7138 appears quite similar to other X-ray sources in the Magellanic Clouds, such as CAL 83, CAL 87, and RX J0527.8-6954, which show only extreme ultrasoft (EUS) X-ray spectra. The discovery of RX J0059.2-7138, a probable high-mass X-ray binary, clearly indicates that EUS spectra may arise from accretion-powered neutron-star X-ray sources. This result lends support to the idea that some of the ''pure'' EUS sources may be shrouded low-mass X-ray binaries rather than accreting white dwarfs.