We present the results of a Goddard High Resolution Spectrograph (GHRS) study of flow and turbulent velocities in the outer atmosphere of the M2 Iab supergiant alpha Ori. Ions with observed chromospheric emission features include C I, C II, Si I, Si II, O I, Co II, Al II, Mg II, Cr II, Fe I, and Fe II, while the photospheric absorption spectrum is dominated by Fe I. The widths of optically thin lines of C II], Si II], Co II, and Fe II indicate an average chromospheric turbulent velocity (Doppler FWHM) in the range of 31-35 km s(-1), which is substantially above the expected chromospheric sound velocity. The shape of the C II] profiles indicates that this turbulence is probably anisotropically distributed, with velocities preferentially directed along and/or perpendicular to the radial direction. The radial velocity of near-UV Fe I photospheric absorption lines averaged 18 km s(-1), somewhat smaller than the systemic RV of alpha Ori itself (21 km s(-1)) and significantly smaller than the RV of the optical Fe I lines (23 km s(-1)) at the time of the HST observations. The various components of the chromospheric emission lines had radial velocities in the range 7-36 km s(-1). The emission wings for most of the ionic species averaged similar to 20 km s(-1), i.e., they were nominally at rest with regard to the star. The central absorption features in these lines, however, showed lower velocities, averaging similar to 16 km s(-1). The radial velocity of the central reversals in the stronger lines was also found to be correlated with the opacity of the lines, changing from similar to 20 km s(-1) for the lower opacity lines to similar to 14 km s(-1) for the higher opacity lines. This implies that we are directly viewing the acceleration of the stellar wind in the chromosphere from rest to about 7 km s(-1). Contrary to the results reported earlier by Carpenter (1984) on the basis of IUE data, there is no indication of a deceleration of the wind at large distances from the star.