Chemically ordered bimetallic nanoparticles are promising candidates for magnetic-storage applications. However, the use of sub-10nm nanomagnets requires further study of possible size effects on their physical properties. Here, the effects of size and morphology on the order-disorder phase transition temperature of CoPt nanoparticles (T(C)(NP)) have been investigated experimentally, using transmission electron microscopy, and theoretically, with canonical Monte Carlo simulations. For 2.4-3-nm particles, T(C)(NP) is found to be 325-175 degrees C lower than the bulk material transition temperature, consistent with our Monte Carlo simulations. Furthermore, we establish that T(C)(NP) is also sensitive to the shape of the nanoparticles, because only one dimension of the particle (that is, in-plane size or thickness) smaller than 3nm is sufficient to induce a considerable depression of T(C)(NP). This work emphasizes the necessity of taking into account the three-dimensional morphology of nano-objects to understand and control their structural properties.