More than half of all stars in the local Universe are found in massive spheroidal galaxies, which are characterized by old stellar populations with little or no current star formation. In present models, such galaxies appear rather late in the history of the Universe as the culmination of a hierarchical merging process, in which larger galaxies are assembled through mergers of smaller precursor galaxies. But observations have not yet established how, or even when, the massive spheroidals formed, nor if their seemingly sudden appearance when the Universe was about half its present age (at redshift z approximately 1) results from a real evolutionary effect (such as a peak of mergers) or from the observational difficulty of identifying them at earlier epochs. Here we report the spectroscopic and morphological identification of four old, fully assembled, massive (10(11) solar masses) spheroidal galaxies at l.6 < z < 1.9, the most distant such objects currently known. The existence of such systems when the Universe was only about one-quarter of its present age shows that the build-up of massive early-type galaxies was much faster in the early Universe than has been expected from theoretical simulations.