[en] When a molecular ion collides with a free electron it can capture the electron and dissociate. The resulting process of Dissociative Recombination (DR) is a process of great significance in a wide variety of plasma environments. In this process, the capture of a free electron leads to the formation of an highly excited state of the neutral molecule, which then dissociates into neutral fragments with kinetic energy and, possibly, internal excitation depending on the energy balance of the reaction. Despite its importance, the DR process is still not yet completely understood theoretically. This is mainly due to the complexity of the nature and dynamics of highly excited molecular states, especially when several channels are involved, as is usually the situation in DR. from experimental point of view, for direct comparison between experiment and theory, this complexity requires detailed experimental data, including the knowledge of both the initial state of the molecular ion, to which DR is very sensitive, and of the final quantum states of the DR products. Inherent un- certainties in the initial vibrational excitation of the laboratory molecular ions was the main drawback of the experiments conducted over the years to, study DR. A substantial progress in the understanding of the DR process was achieved with the introduction (about five years ago) of a new experimental approach, which uses heavy-ion storage ring technique. In a storage ring, one can store many molecular ions for a time which is long enough to allow complete radiative deexcitation of tile initial electronic and vibrational excitation coming from the ion source. Those vibrationally cold ions are then merged with an intense electron beam to measure their DR cross section. Further experimental progress was the inclusion of two and three-dimensional molecular imaging techniques [1] for the measurement of the branching ratio to different final quantum states of the neutral DR fragments. This talk will present results of the next step forward of this method. Rate coefficients and branching ratios were obtained for DR of HD⁺ in selected initial vibrational quantum states (ν=0-7), and not only in the ground vibrational state. The experiment was carried out at the Test Storage Ring (TSR) located at the Max-Planck-Institut fur Kernphysik, Heidelberg with a 2-MeV HD⁺ beam. The (initial)-state-to-(final)-state relative DR reaction rates were measured using two-dimensional imaging technique for the detection of the DR fragments. These rates were then converted to rate coefficients using the vibrational population distribution of the stored ion beam, as measured by Coulomb explosion imaging (CEI) of molecular ions which were extracted from the ring. Both measurements were done ac a function of storage time, during thr radiative relaxation of the HD⁺ molecular ions. Present theoretical predictions are in discrepancy with the dta