We present an optically-based study of the alignment between the radio axes and the optical major axes of eight z approximately 0.7 radio galaxies in a 7C sample. The radio galaxies in this sample are approximately 20-times less radio luminous than 3C galaxies at the same redshift, and are significantly less radio-luminous than any other well-defined samples studied to date. Using Nordic Optical Telescope images taken in good seeing conditions at rest-frame wavelengths just longward of the 4000A break, we find a statistically significant alignment effect in the 7C sample. Furthermore, in two cases where the aligned components are well separated from the host we have been able to confirm spectroscopically that they are indeed at the same redshift as the radio galaxy. However, a quantitative analysis of the alignment in this sample and in a corresponding 3C sample from HST (Hubble Space Telescope) archival data indicates that the percentage of aligned flux may be lower and of smaller spatial scale in the 7C sample. Our study suggests that alignments on the 50-kpc scale are probably closely related to the radio luminosity, whereas those on the 15 kpc scale are not. We discuss these results in the context of popular models for the alignment effect.