The metal-to-insulator transition in rutile VO_{2} has proven uniquely difficult to characterize because of the complex interplay between electron correlations and atomic structure. Here, we report the discovery of the sudden collapse of three-dimensional order in the low-temperature phase of V_{1-x}Mo_{x}O_{2} at x=0.17 and the emergence of a novel frustrated two-dimensional order at x=0.19, with only a slight change in electronic properties. Single crystal diffuse x-ray scattering reveals that this transition from the 3D M1 phase to a 2D variant of the M2 phase results in long-range structural correlations along symmetry-equivalent (11L) planes of the tetragonal rutile structure, yet extremely short-range correlations transverse to these planes. These findings suggest that this two dimensionality results from a novel form of geometric frustration that is essentially structural in origin.