CoPt and Co₃Pt alloy epitaxial films with the close-packed plane parallel to the surface are prepared on MgO(111) single-crystal substrates heated at 300 °C by varying the film thickness in a range between 2 and 40 nm. The structure is investigated from the viewpoints of the order degree and the atomic stacking sequence of close-packed plane. Metastable ordered phase formation is recognized for both CoPt and Co₃Pt films. The CoPt and the Co₃Pt films involve L1₁ and B_h(+D0₁₉) phases, respectively. Higher order degrees are recognized for thicker CoPt and Co₃Pt films. As the thickness increases, the strain caused by accommodation of the lattice mismatch between film and substrate decreases. The ratios of out-of-plane to in-plane lattice spacings, 2d_L11(222) / 2d_L11(440) and 2d_{Bh(+D019)(0002)} / 2d_{Bh(+D019)(2240)}, of CoPt and Co₃Pt films, respectively, decrease from 1.638 to 1.630 and from 1.637 to 1.616 with increasing the thickness from 2 to 40 nm. The films involving metastable phases show strong perpendicular magnetic anisotropies reflecting the magnetocrystalline anisotropies of L1₁ and B_h(+D0₁₉) crystals.