We analyze the weak-segregation, thermodynamic predictions of a simple mean-field model of diblock copolymer melts sandwiched between surfaces that interact weakly with melts. Under many conditions, these films do not exhibit a phase transition although the segregation that develops within them resembles ordering in the bulk. When transitions occur, they result from a competition between surface and bulk effects and have no bulk analog. Thin films confined by identical surfaces and films bounded by surfaces having equal in magnitude, but opposite affinities for the monomers may exhibit first- as well as second-order intrafilm transitions when the melt-surface interactions are small. The model predicts that second-order transitions do not occur in films with large surface affinities and that films bounded by different surfaces undergo only first-order intrafilm transitions. The instabilities that lead to the second-order transitions in films with symmetry-breaking affinities disappear with increases in the film thickness.