The energy-resolved rate coefficient for the dissociative recombination (DR) of H(3)(+) with slow electrons has been measured by the storage-ring method using an ion beam produced from a radiofrequency multipole ion trap, employing buffer-gas cooling at 13 K. The electron energy spread of the merged-beams measurement is reduced to 500 microeV by using a cryogenic GaAs photocathode. This and a previous cold- measurement jointly confirm the capability of ion storage rings, with suitable ion sources, to store and investigate H(3)(+) in the two lowest, (J,G) = (1,1) and (1,0) rotational states prevailing also in cold interstellar matter. The use of para-H(2) in the ion source, expected to enhance para-H(3)(+) in the stored ion beam, is found to increase the DR rate coefficient at meV electron energies.