A kinetic theory is used to find the dispersion relation of a plasma-loaded free-electron laser that takes into account the velocity spread of both beam and plasma electrons. For the weak-coupling regime, the dispersion relation for a cold electron beam and plasma is obtained and is studied numerically. The kinetic theory predicts that, for the cold beam and plasma, plasma density increases the growth rate of the higher-frequency resonance and lowers it for the lower-frequency resonance up to a certain plasma density. By increasing the plasma density further, a critical density will be reached that diminishes the instability.