Low-threshold optical bistability is essential for practical nonlinear optical applications. Many bistable optical devices based on high-quality-factor resonators have been proposed to reduce the threshold intensity. However, demonstrating high-quality-factor resonance requires complex fabrication techniques. In this work, we numerically demonstrate optical bistability with bound states in the continuum in a simple one-dimensional Si photonic crystal. The designed structure supports bound states in the continuum, producing an ultrahigh quality factor without tough fabrication conditions. The threshold intensity of the designed device is 150 MW/cm2 at the optical communication wavelength. This scheme may lead to a new class of nonlinear photonics.