Context: The high degree of deuteration observed in some prestellar cores depends on the ortho-to-para H2 ratio through the H3+ fractionation. Aims: We want to constrain the ortho/para H2 ratio across the L183 prestellar core. This is required to correctly describe the deuteration amplification phenomenon in depleted cores such as L183 and to relate the total (ortho+para) H2D+ abundance to the sole ortho-H2D+ column density measurement. Methods: To constrain this ortho/para H2 ratio and derive its profile, we make use of the N2D^+/N2H+ ratio and of the ortho-H2D+ observations performed across the prestellar core. We use two simple chemical models limited to an almost totally depleted core description. New dissociative recombination and trihydrogen cation-dihydrogen reaction rates (including all isotopologues) are presented in this paper and included in our models. Results: We estimate the H2D+ ortho/para ratio in the L183 cloud, and constrain the H2 ortho/para ratio: we show that it varies across the prestellar core by at least an order of magnitude, being still very high (≈0.1) in most of the cloud. Our time-dependent model indicates that the prestellar core is presumably older than 1.5-2 × 105 years but that it may not be much older. We also show that it has reached its present density only recently and that its contraction from a uniform density cloud can be constrained. Conclusions: A proper understanding of deuteration chemistry cannot be attained without taking into account the whole ortho/para family of molecular hydrogen and trihydrogen cation isotopologues as their relations are of utmost importance in the global scheme. Tracing the ortho/para H2 ratio should also place useful constraints on the dynamical evolution of prestellar cores. Appendices A and B are only available in electronic form at http://www.aanda.org< Réduire