We study the electron spin resonance line-width (ESRLW) of $Fe^{3+}$ in crystalline $LiNbO_3$ ; the ESRLW is obtained using the projection operator method (POM) developed by Argyres and Sigel. The ESRLW is calculated to be axially symmetric about the c-axis and is analyzed by the spin Hamiltonian with an isotopic g factor at a frequency of 9.5 GHz. The ESRLW increases exponentially as the temperature increases, and the ESRLW is almost constant in the high-temperature region (T>8000 K). This kind of temperature dependence of the ESRLW indicates a motional narrowing of the spectrum when $Fe^{3+}$ ions substitute the $Nb^{5+}$ ions in an off-center position. It is clear from this feature that there are two different regions in the graph of the temperature dependence of the ESRLW.