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Hubble Space Telescope images of high-redshift galaxies selected via color and photometric redshifts are used to examine the size and axial-ratio distribution of galaxies as a function of redshift at lookback times t > 8 Gyr. These parameters are measured at rest-frame UV wavelengths (1200-2000 Angstroms) on images with a rest-frame resolution of less than 0.8 kpc. Galaxy radii are found to scale with redshift approximately as the inverse of the Hubble parameter H(z). This is in accord with the theoretical expectation that the typical sizes of the luminous parts of galaxies should track the expected evolution in the virial radius of dark-matter halos. The mean ratio of semi-major to semi-minor axis for a bright well-resolved sample of galaxies at z ~ 4 is b/a = 0.65, suggesting that these Lyman break galaxies are not drawn from a spheroidal population. However the median concentration index of this sample is C = 3.5, which is closer to the typical concentration indices, C ~ 4, of nearby elliptical galaxies than to the values, C<2 for local disk galaxies of type Sb and later.