The incommensurately modulated structure of a square Te-net, namely that of K1/3Ba2/3AgTe2, is determined from single-crystal X-ray diffraction data within a (3+1)D higher dimension formalism. The phase is shown to crystallize in the monoclinic symmetry, P2(1)(alpha 0 gamma) superspace group with the following lattice parameters: a = 4.6441(10) A, b = 4.6292(12) A, c = 23.765(9) A, and beta = 101.28(2) degrees with q = 0.3248(6)a* -0.071(8)c*, that is, in a symmetry different from that reported for the average structure (tetragonal) or that assumed from electron diffraction measurements (orthorhombic). After the introduction of a crenel function for the Te displacive description, the refinement converged to a residual factor R = 0.033 for 2583 observed reflections and 115 parameters (R = 0.024 and 0.101 for 1925 main reflections and 658 first-order satellites, respectively). The [Ag2-Te2] and the Ba/K layers are found to be only weakly modulated. The modulation of the square Te-net is, however, both substantial and unique. Namely, it results in two different units: a "V"-shaped Te3 trimer and a "W"-shaped Te5 pentamer. To examine both unit types, which are segregated in domains that aperiodically alternate within the Te layers, first principles electronic band structure calculations were carried out for three model commensurate structures using the tight-binding linear-muffin-tin-orbital method (LMTO). The calculations show that the distorted structures of V-pattern (model 2) and W-pattern (model 3) are more stable than the average structure (model 1) and that the V-pattern distortion provides a slightly larger stabilization than does the W-pattern distortion. The Fermi surface calculated for the average structure shows nesting vectors that are consistent with the occurrence of the V- and W-pattern distortions in the Te layers. However, these vectors do not predict the observed modulation vector of the incommensurately distorted structure, because the stabilization energy associated with the distortion is not mainly dominated by the energy lowering of the occupied band levels near the Fermi level.