[ $^1H$ ] nuclear magnetic resonance (NMR) experiments have been performed at 30 - 300 K and 7 T to investigate dynamics of hydrogen bond network in the single crystal $(NH_4)_3H(SO_4)_2$. The two proton sites, ammonium proton and hydrogen-bond proton, are identified from the $^1H$ NMR MAS spectrum at 340 K. As temperature decreases, the $^1H$ NMR spectrum shifts to the higher frequency side with a larger linewidth. The spectrum at 65 K shows a distinctive change in line shape toward the ferroelectric transition at 63 K. The measured values of $T_1$ for ammonium and hydrogen-bond protons are similar in the whole range of temperature. $T_1$ of $^1H$ NMR shows a gradual decrease down to 120 K and starts to steeply increase below 100 K. Then $T_1$ shows abrupt decrease below 70 K with a sharp minimum at 63 K, where the ferroelectric transition occurs. This temperature dependence of spectrum and $T_1$ clearly prove that the large change in the dynamics of hydrogen bond network is associated with the ferroelectric phase transition at 63 K.