H3+ plays a key role in interstellar chemistry as the initiator of ion-molecule chemistry. The amount of H3+ observed in dense interstellar clouds is consistent with expectations, but the large abundance of H3+ seen in diffuse clouds is not easily explained by simple chemical models. A crucial parameter in predicting the abundance of H3+ in diffuse clouds is the rate constant for dissociative recombination (DR) with electrons. The value of this constant has been very controversial, because different experimental techniques have yielded very different results, perhaps owing to varying degrees of rotational and vibrational excitation of the H3+ ions. If the value of this rate constant under interstellar conditions were much lower than usually assumed, the large H3+ abundance could be easily explained. In an attempt to pin down this crucial rate constant, we have performed DR measurements at the CRYRING ion storage ring in Stockholm, using a supersonic expansion ion source to produce rotationally cold H3+ ions. These measurements suggest that the DR rate constant in diffuse clouds is not much lower than usually assumed and that the abundant H3+ must be due to either a low electron fraction or a high ionization rate.