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This paper studies a downlink secure integrated sensing and communication (ISAC) system, in which a multi-antenna base station (BS) transmits confidential messages to a single-antenna communication user (CU) while performing sensing on targets that may act as suspicious eavesdroppers. To ensure the quality of target sensing while preventing their potential eavesdropping, the BS combines the transmit confidential information signals with additional dedicated sensing signals, which play a dual role of artificial noise (AN) for degrading the qualities of eavesdropping channels. Under this setup, we jointly design the transmit information and sensing beamforming, with the objective of minimizing the weighted sum of beampattern matching errors and cross-correlation patterns for sensing subject to secure communication constraints. The robust design takes into account the channel state information (CSI) imperfectness of the eavesdroppers in two practical CSI error scenarios. First, we consider the scenario with bounded CSI errors of eavesdroppers, in which the worst-case secrecy rate constraint is adopted to ensure secure communication performance. In this scenario, we present the optimal solution to the worst-case secrecy rate constrained sensing beampattern optimization problem, by adopting the techniques of S-procedure, semi-definite relaxation (SDR), and a one-dimensional (1D) search, for which the tightness of the SDR is rigorously proved. Next, we consider the scenario with Gaussian CSI errors of eavesdroppers, in which the secrecy outage probability constraint is adopted. In this scenario, we present an efficient algorithm to solve the more challenging secrecy outage-constrained sensing beampattern optimization problem, by exploiting the convex restriction technique based on the Bernstein-type inequality, together with the SDR and 1D search.