We describe the first measurement of an electron beam size in the accelerator test facility damping ring at KEK with a laser wire beam profile monitor. This monitor is based upon the Compton scattering process of electrons with a laser light target, which is produced by injecting a cw laser beam into a Fabry-Pérot optical cavity. We have observed clear signals of the Compton scattered photons and confirmed that the observed energy spectrum as well as the count rate agree with the expected ones. From the measurement, we have deduced the vertical beam size σ_{b} to be 9.8±1.1±0.4 μm, where the first (second) error represents statistical (systematic) uncertainty. Various improvements are in progress to enhance the signal-to-noise ratio, which is essential for the detailed study of the beam dynamics.