We explore an alternative way to fabricate (In,Ga)N/GaN short-period superlattices on GaN(0001) by plasma-assisted molecular beam epitaxy. We exploit the existence of an In adsorbate structure manifesting itself by a $(sqrt{3}times!sqrt{3})text{R}30^{circ}$ surface reconstruction observed in-situ by reflection high-energy electron diffraction. This In adlayer accommodates a maximum of 1/3 monolayer of In on the GaN surface and, under suitable conditions, can be embedded into GaN to form an In$_{0.33}$Ga$_{0.67}$N quantum sheet whose width is naturally limited to a single monolayer. Periodically inserting these quantum sheets, we synthesize (In,Ga)N/GaN short-period superlattices with abrupt interfaces and high periodicity as demonstrated by x-ray diffractometry and scanning transmission electron microscopy. The embedded quantum sheets are found to consist of single monolayers with an In content of 0.25-0.29. For a barrier thickness of 6 monolayers, the superlattice gives rise to a photoluminescence band at 3.16 eV, close to the theoretically predicted values for these structures.