Interferometric wavelength meters have attained frequency resolutions down to the megahertz range. In particular, Fizeau interferometers, which have no moving parts, are becoming a popular tool for laser characterization and stabilization. In this paper, we characterize such a wavelength meter using an ultrastable laser in terms of relative frequency instability σ_y(τ) and demonstrate that it can achieve a short-term instability σ_y(1s) ≈ 2×10^-10 and a frequency drift of order 10 MHz/day. We use this apparatus to demonstrate frequency control of a near-infrared laser, where a frequency instability below 3×10^-10 from 1 to 2000 s is achieved. Such performance is, for example, adequate for ion trapping and atom cooling experiments.