We have used the Goddard High-Resolution Spectrograph (GHRS) on the Hubble Space Telescope to study the wind of the hybrid-chromosphere star alpha TrA (K4 II). The stellar wind produces significant absorption at negative radial velocities in the chromospheric Mg II resonance lines (h and k). Spectra obtained with the GHRS echelle high-resolution grating (lambda/Delta lambda approximate to 85,000) on 1993 February 10 and 1994 May 1 reveal complex interstellar absorption in the Mg II emission lines and a high-velocity wind absorption feature centered near -95 km s(-1). The 1993 February observation shows an asymmetry of the Mg II emission cores, corresponding to an apparent redshift of 6.0 +/- 1.5 km s(-1). We construct a simple wind model that explains several of the key observational features. The scattering of the Mg II h and k photons in a geometrically extended region dominates the observed flux near line center, which supports the assignment of the low-velocity absorption components to interstellar absorption rather than to a chromospheric self-reversal. For the 1993 February observation, the parameters for our simple wind model are as follows: terminal velocity V(infinity) = 100 km s(-1) turbulent velocity V-turb = 24 km s(-1), M similar to 1.8 x 10-(10) M. yr(-1), for a fixed value of the velocity-law parameter beta = 1 and fixed stellar radius of R* = 97 R., assuming Mg II is the dominant ionization state in the flow. Our analysis of the 1994 May observation resulted in similar values for these parameters, and the mass-loss rate could be as small as M similar to 1.6 x 10-(10) M. yr(-1). The value of beta is uncertain (greater than or equal to 0.3) and if beta similar to 3.5 as found from the recent analysis of the zeta Aurigae systems, M could be larger by a factor of 3-4. A comparison of our result with numerical solutions to the momentum and conservation equations reveals that the derived velocity distribution lies within a limited region of parameter space where there is a large nonthermal pressure on the plasma close to the base of the wind consistent with previous wind models for alpha TrA. Our best model fit to the two interstellar absorption components indicates a total hydrogen column density toward alpha TrA of N-HI = 2 x 10(19) cm(-2).