Integrated sensing and communication (ISAC), and intelligent reflecting surface (IRS) are envisioned as revolutionary technologies to enhance spectral and energy efficiencies for next wireless system generations. For the first time, this paper focuses on the channel estimation problem in an IRS-assisted ISAC system. This problem is challenging due to the lack of signal processing capacity in passive IRS, as well as the presence of mutual interference between sensing and communication (SAC) signals in ISAC systems. A three-stage approach is proposed to decouple the estimation problem into sub-ones, including the estimation of the direct SAC channels in the first stage, reflected communication channel in the second stage, and reflected sensing channel in the third stage. The proposed three-stage approach is based on a deep-learning framework, which involves two different convolutional neural network (CNN) architectures to estimate the channels at the full-duplex ISAC base station. Furthermore, two types of input-output pairs to train the CNNs are carefully designed, which affect the estimation performance under various signal-to-noise ratio conditions and system parameters. Simulation results validate the superiority of the proposed estimation approach compared to the least-squares baseline scheme, and its computational complexity is also analyzed.