We derive asymptotically exact formulas for the equilibrium magnetic stripe period in ultrathin films with out-of-plane anisotropy that include the full domain wall magnetic dipolar energy. Starting with the reduced two-dimensional micromagnetic model valid for thin films, we obtain the leading order approximation for the energy per unit volume in the vanishing film thickness limit in the case of Bloch and N\'eel wall rotations. Its minimization in the stripe period leads to an analytical expression for the equilibrium period with a prefactor proportional to the Bloch wall width. The constant in the prefactor, related to the long-range dipolar interactions, is carefully evaluated. This results in a remarkable agreement of the stripe domain energy density and stripe period predicted by our analytical formulas with micromagnetic simulations. Our formula can be used to accurately deduce magnetic parameters from the experimental measurements of the stripe period and to systematically predict the equilibrium stripe periods in ultrathin films.