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In this paper, an intelligent reflecting surface (IRS) is leveraged to enhance the physical layer security of an integrated sensing and communication (ISAC) system in which the IRS is deployed to not only assist the downlink communication for multiple users, but also create a virtual line-of-sight (LoS) link for target sensing. In particular, we consider a challenging scenario where the target may be a suspicious eavesdropper that potentially intercepts the communication-user information transmitted by the base station (BS). We investigate the joint design of the phase shifts at the IRS and the communication as well as radar beamformers at the BS to maximize the sensing beampattern gain towards the target, subject to the maximum information leakage to the eavesdropping target and the minimum signal-to-interference-plus-noise ratio (SINR) required by users. Based on the availability of perfect channel state information (CSI) of all involved user links and the accurate target location at the BS, two scenarios are considered and two different optimization algorithms are proposed. For the ideal scenario where the CSI of the user links and the target location are perfectly known at the BS, a penalty-based algorithm is proposed to obtain a high-quality solution. In particular, the beamformers are obtained with a semi-closed-form solution using Lagrange duality and the IRS phase shifts are solved for in closed form by applying the majorization-minimization (MM) method. On the other hand, for the more practical scenario where the CSI is imperfect and the target location is uncertain, a robust algorithm based on the $\cal S$-procedure and sign-definiteness approaches is proposed. Simulation results demonstrate the effectiveness of the proposed scheme in achieving a trade-off between the communication quality and the sensing quality.