DS Tuc Ab is a Neptune-sized planet that orbits around a member of the 45 Myr old Tucana-Horologium moving group. Here, we report the measurement of the sky-projected angle between the stellar spin axis and the planets orbital axis, based on the observation of the Rossiter-McLaughlin effect during three separate planetary transits. The orbit appears to be well aligned with the equator of the host star, with a projected obliquity of lambda = 2.5 +1.0/-0.9 deg. In addition to the distortions in the stellar absorption lines due to the transiting planet, we observed variations that we attribute to large starspots, with angular sizes of tens of degrees. The technique we have developed for simultaneous modeling of starspots and the planet-induced distortions may be useful in other observations of planets around active stars.