We demonstrate the stray-field-mediated skyrmion stabilizing capabilities of ultrathin exchange-decoupled antisymmetric ferromagnetic bilayers based on conventional transition metal materials. Using an asymptotically exact micromagnetic model valid in the ultrathin film limit, we show that the antisymmetric tailoring of the bilayer allows the Dzyaloshinskii-Moriya interaction and the dipolar interaction to act synergistically to stabilize skyrmions, in contrast to the monolayer case, in which these energies compete. To obtain optimal stability of these skyrmions against collapse and bursting – the two fundamental processes determining skyrmion lifetime, we carry out an asymptotic analysis of the saddle point solution that separates the skyrmion from the demagnetized state. The result is an optimal stability line for compact skyrmions in the non-dimensional parameter space of the effective Dzyaloshinskii-Moriya interaction strength and the effective film thickness. Our predictions are confirmed by extensive micromagnetic simulations of antisymmetric bilayers, using magnetic parameters of the conventional Pt/Co/AlO${}_x$ systems. Our results may provide a new pathway for experimental observations of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications.