This thesis presents work on merged-beams experiments using the DESIREE facility, which consists of two electrostatic ion-storage rings operated at cryogenic temperatures of about 13 K. The two storage rings have a common straight section, where oppositely charged ions can interact at sub-eV collision energies. The work consists of development of the experimental method, tools for data analysis, and simulations for interpretation of the experimental results. This development has led to the first results from merged-beams experiments at DESIREE. The experiments are performed on mutual neutralization, a collision process where an electron is transferred from a negatively charged ion to a positively charged ion, resulting in two neutral products. This reaction is investigated for three systems: Li⁺ with D⁻, Na⁺ with D⁻, and Mg⁺ with D⁻. The reactions studied are of astrophysical interest, in particular for modeling of stellar atmospheres. The detection of the reaction products is done by using a position- and time-sensitive detector, and from this information it is possible to determine the quantum states involved in the reaction. The branching fractions of the reaction channels are measured and compared to theoretical modeling.