Pluto and Eris are icy dwarf planets with nearly identical sizes, comparable densities and similar surface compositions as revealed by spectroscopic studies. Pluto possesses an atmosphere whereas Eris does not; the difference probably arises from their differing distances from the Sun, and explains their different albedos. Makemake is another icy dwarf planet with a spectrum similar to Eris and Pluto, and is currently at a distance to the Sun intermediate between the two. Although Makemake's size (1,420 +/- 60 km) and albedo are roughly known, there has been no constraint on its density and there were expectations that it could have a Pluto-like atmosphere. Here we report the results from a stellar occultation by Makemake on 2011 April 23. Our preferred solution that fits the occultation chords corresponds to a body with projected axes of 1,430 +/- 9 km (1σ) and 1,502 +/- 45 km, implying a V-band geometric albedo pV = 0.77 +/- 0.03. This albedo is larger than that of Pluto, but smaller than that of Eris. The disappearances and reappearances of the star were abrupt, showing that Makemake has no global Pluto-like atmosphere at an upper limit of 4-12 nanobar (1σ) for the surface pressure, although a localized atmosphere is possible. A density of 1.7 +/- 0.3 g cm-3 is inferred from the data.