This paper is the first report of robotic intracytoplasmic sperm injection (ICSI). ICSI is a clinical procedure performed worldwide in fertility clinics, requiring pick-up of a single sperm and insertion of it into an oocyte (i.e., egg cell). Since its invention 20 years ago, ICSI has been conducted manually by a handful of highly skilled embryologists; however, success rates vary significantly among clinics due to poor reproducibility and inconsistency across operators. We leverage our work in robotic cell injection to realize robotic ICSI and aim ultimately, to standardize how clinical ICSI is performed. This paper presents some of the technical aspects of our robotic ICSI system, including a cell holding device, motion control, and computer vision algorithms. The system performs visual tracking of single sperm, robotic immobilization of sperm, aspiration of sperm with picoliter volume, and insertion of sperm into an oocyte with a high degree of reproducibility. The system requires minimal human involvement (requiring only a few computer mouse clicks), and is human operator skill independent. Using the hamster oocyte-human sperm model in preliminary trials, the robotic system demonstrated a high success rate of 90.0% and survival rate of 90.7% (n=120).