The determination of the dissociative recombination rate coefficient of H(3) (+) has had a turbulent history, but both experiment and theory have recently converged to a common value. Despite this convergence, it has not been clear if there should be a difference between the rate coefficients for ortho-H(3) (+) and para-H(3) (+). A difference has been predicted theoretically and could conceivably impact the ortho:para ratio of H(3) (+) in the diffuse interstellar medium, where H(3) (+) has been widely observed. We present the results of an experiment at the CRYRING ion storage ring in which we investigated the dissociative recombination of highly enriched ( approximately 83.6%) para-H(3) (+) using a supersonic expansion source that produced ions with T(rot) approximately 60-100 K. We observed an increase in the low energy recombination rate coefficient of the enriched para-H(3) (+) by a factor of approximately 1.25 in comparison to H(3) (+) produced from normal H(2) (ortho:para=3:1). The ratio of the rate coefficients of pure para-H(3) (+) to that of pure ortho-H(3) (+) is inferred to be approximately 2 at low collision energies; the corresponding ratio of the thermal rate coefficients is approximately 1.5 at electron temperatures from 60 to 1000 K. We conclude that this difference is unlikely to have an impact on the interstellar ortho:para ratio of H(3) (+).