We have investigated dissociative recombination (DR) of NH$^+$ with electrons using a merged beams configuration at the TSR heavy-ion storage ring located at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. We present our measured absolute merged beams recombination rate coefficient for collision energies from 0 to 12 eV. From these data we have extracted a cross section which we have transformed to a plasma rate coefficient for the collisional plasma temperature range from $T_{rm pl} = 10$ to $18000$ K. We show that the NH$^+$ DR rate coefficient data in current astrochemical models are underestimated by up to a factor of $sim 9$. Our new data will result in predicted NH$^+$ abundances lower than calculated by present models. This is in agreement with the sensitivity limits of all observations attempting to detect NH$^+$ in interstellar clouds.