Many computational models of normal vernier acuity make predictions based on the just-noticeable contrast difference. Recently, Hu, Klein and Carney [(1993) Vision Research, 33, 1241-1258] compared vernier acuity and contrast discrimination (jnd) in normal foveal viewing using cosine gratings. In the jnd stimulus the test grating was added in-phase to the (sinusoidal) pedestal, whereas in the vernier stimulus the same test grating was added with an approx. 90 deg phase shift to the pedestal. In the present experiments, we measured thresholds for discriminating changes in relative position and changes in relative contrast for abutting, horizontal cosine gratings in a group of amblyopes using the Hu et al., test-pedestal approach. The approach here is to ask whether the reduced vernier acuity of amblyopes can be understood on the basis of reduced contrast sensitivity or contrast discrimination. Our results show that (i) abutting cosine vernier acuity is strongly dependent on stimulus contrast. (ii) In both anisometropic and strabismic amblyopes, abutting cosine vernier discrimination thresholds are elevated at all contrast levels, even after accounting for reduced target visibility, or contrast discrimination. (iii) For both strabismic and anisometropic amblyopes, the vernier Weber fraction is markedly degraded, while the contrast Weber fraction is normal or nearly so. (iv) In anisometropic amblyopes the elevated vernier thresholds are consistent with the observers' reduced cutoff spatial frequency, i.e. the loss can be accounted for on the basis of a shift in spatial scale. (v) In strabismic amblyopes and in the normal periphery, there appears to be an extra loss, which can not be accounted for by either reduced contrast sensitivity and contrast discrimination or by a shift in spatial scale. (vi) This extra loss cannot be quantitatively mimicked by "undersampling" the stimulus. (vii) Surprisingly, in some strabismics, and in the periphery, at relatively high spatial frequencies, vernier thresholds appear to lose their contrast dependence, suggesting the possibility that there may be qualitative differences between the normal fovea and these degraded visual systems. (viii) This contrast saturation can be mimicked by "undersampling" the target, or by introducing strips of mean luminance between the two vernier gratings, thus mimicking a "scotoma". Taken together with the preceding paper, our results suggest that the extra loss in position acuity of strabismic amblyopes and the normal periphery may be a consequence of noise at a second stage of processing, which selectively degrades position but not contrast discrimination.