Partial-thickness corneal transplants using a deep anterior lamellar keratoplasty (DALK) approach has demonstrated better patient outcomes than a full-thickness cornea transplant. However, despite better clinical outcomes from the DALK procedure, adoption of the technique has been limited because the accurate insertion of the needle into the deep stroma remains technically challenging. In this work, we present a novel hands-free eye mountable robot for automatic needle placement in the cornea, AutoDALK, that has the potential to simplify this critical step in the DALK procedure. The system integrates dual light-weight linear piezo motors, an OCT A-scan distance sensor, and a vacuum trephine-inspired design to enable the safe, consistent, and controllable insertion of a needle into the cornea for the pneumodissection of the anterior cornea from the deep posterior cornea and Descemet's membrane. AutoDALK was designed with feedback from expert corneal surgeons and performance was evaluated by finite element analysis simulation, benchtop testing, and ex vivo experiments to demonstrate the feasibility of the system for clinical applications. The mean open-loop positional deviation was 9.39 µm, while the system repeatability and accuracy were 39.48 µm and 43.18 µm, respectively. The maximum combined thrust of the system was found to be 1.72 N, which exceeds the clinical penetration force of the cornea. In a head-to-head ex vivo comparison against an expert surgeon using a freehand approach, AutoDALK achieved more consistent needle depth, which resulted in fewer perforations of Descemet's membrane and significantly deeper pneumodissection of the stromal tissue. The results of this study indicate that robotic needle insertion has the potential to simplify the most challenging task of the DALK procedure, enable more consistent surgical outcomes for patients, and standardize partial-thickness corneal transplants as the gold standard of care if demonstrated to be more safe and more effective than penetrating keratoplasty.