Transverse interferometric measurements are presented of the plasma channel formed in a hydrogen-filled capillary discharge waveguide recently used to generate 1 GeV electrons in a laser-driven plasma accelerator for the first time. The measurements were found to be in good agreement with nonlocal thermal equilibrium simulations, but showed significant differences with the results of a quasistatic model developed by Bobrova et al. [Phys. Rev. E. 65, 016407 (2001)]. The measurements are used to determine scaling laws for the axial electron density and matched spot size of the plasma channel, enabling optimization of the channel to specific applications.