We have produced a sample of 58 edge-on spiral galaxies at redshifts z ~ 1 selected in the Hubble Ultra Deep Field. For all galaxies we have analyzed the 2D brightness distributions in the V₆₀₆ and i₇₇₅ filters and measured the radial (h_r) and vertical (h_z) exponential scale lengths of the brightness distribution. We have obtained evidence that the relative thickness of the disks of distant galaxies, i.e., the ratio of the vertical and radial scale lengths, on average, exceeds the relative thickness of the disks of nearby spiral galaxies. The vertical scale length h_z of the stellar disks of galaxies shows no big changes at z = 1. The possibility of the evolution of the radial scale length h_z for the brightness distribution with redshift is discussed.