Contact-line pinning is a fundamental limitation to the motion of contact lines of liquids on solid surfaces. When a sessile droplet evaporates, contact-line pinning typically results in either a stick-slip evaporation mode, where the contact line pins and depins from the surface in an uncontrolled manner, or a constant contact-area mode with a pinned contact line. Pinning prevents the observation of the quasi-equilibrium constant contact-angle mode of evaporation, which has never been observed for sessile droplets of water directly resting on a smooth, nontextured, solid surface. Here, we report the evaporation of a sessile droplet from a flat glass substrate treated with a smooth, slippery, omniphobic covalently attached liquid-like coating. Our characterization of the surfaces shows high contact line mobility with an extremely low contact angle hysteresis of similar to 1 degrees and reveals a step change in the value of the contact angle from 101 degrees to 105 degrees between a relative humidity (RH) of 30 and 40%, in a manner reminiscent of the transition observed in a type V adsorption isotherm. We observe the evaporation of small sessile droplets in a chamber held at a constant temperature, T = (25.0 +/- 0.1) degrees C and at constant RH across the range RH = 10-70%. In all cases, a constant contact-angle mode of evaporation is observed for most of the evaporation time. Furthermore, we analyze the evaporation sequences using the Picknett and Bexon ideal constant contact-angle mode for diffusion-limited evaporation. The resulting estimate for the diffusion coefficient, D-E, of water vapor in air of D-E = (2.44 +/- 0.48) X 10(-5) m(2) s(-1) is accurate to within 2% of the value reported in the literature, thus validating the constant contact-angle mode of the diffusion-limited evaporation model.