Rate coefficients for dissociative recombination of HD+ in selected vibrational states have been measured by a combination of two molecular fragment imaging methods by using the heavy-ion storage ring technique. Recombination fragment imaging yields state-to-state reaction rates. These rates are converted to rate coefficients by using vibrational level populations of the stored ion beam, derived from nuclear coordinate distributions measured on extracted ions. The results show strongly increasing rate coefficients for high vibrational excitation, where additional dissociation routes open up, in agreement with a theoretical calculation. Very low rate coefficients are found for certain, isolated vibrational states.