We present the first spectroscopic measurements of the shape of the far-ultraviolet (far-UV; lambda = 950-1500 A) dust attenuation curve at high redshift (z~ 3). Our analysis employs rest-frame UV spectra of 933 galaxies at z~ 3, 121 of which have very deep spectroscopic observations (> ~7 hr) at lambda = 850-1300 A, with the Low Resolution Imaging Spectrograph on the Keck Telescope. By using an iterative approach in which we calculate the ratios of composite spectra in different bins of continuum color excess, E(B- V), we derive a dust curve that implies a lower attenuation in the far-UV for a given E(B- V) than those obtained with standard attenuation curves. We demonstrate that the UV composite spectra of z~ 3 galaxies can be modeled well by assuming our new attenuation curve, a high covering fraction of H I, and absorption from the Lyman-Werner bands of H sub(2) with a small (< ~20%) covering fraction. The low covering fraction of H sub(2) relative to that of the H sub(1) and dust suggests that most of the dust in the ISM of typical galaxies at z~ 3 is unrelated to the catalysis of H sub(2), and is associated with other phases of the ISM (i.e., the ionized and neutral gas). The far-UV dust curve implies a factor of thickapproximate2 lower dust attenuation of Lyman continuum (ionizing) photons relative to those inferred from the most commonly assumed attenuation curves for L* galaxies at z~ 3. Our results may be utilized to assess the degree to which ionizing photons are attenuated in H II regions or, more generally, in the ionized or low column density (N(H I) < ~ 10 super(17,2) cm super(-2)) neutral ISM of high-redshift galaxies.