This paper considers an artificial noise (AN) aided secrecy rate maximization (SRM) problem for a multi-input single-output (MISO) channel overheard by multiple single-antenna eavesdroppers. We assume that the transmitter has perfect knowledge about the channel to the desired user but imperfect knowledge about the channels to the eavesdroppers. Therefore, the resultant SRM problem is formulated in the way that we maximize the worst-case secrecy rate by jointly designing the signal covariance W and the AN covariance Σ. However, such a worst-case SRM problem turns out to be hard to optimize, since it is nonconvex in W and Σ jointly. Moreover, it falls into the class of semi-infinite optimization problems. Through a careful reformulation, we show that the worst-case SRM problem can be handled by performing a one-dimensional line search in which a sequence of semidefinite programs (SDPs) are involved. Moreover, we also show that the optimal W admits a rank-one structure, implying that transmit beamforming is secrecy rate optimal under the considered scenario. Simulation results are provided to demonstrate the robustness and effectiveness of the proposed design compared to a non-robust AN design.