We propose and implement a geodesic half-Maxwell fish-eye (MFE)-lens antenna. The lens was optimized using an in-house physical optics (PO) code adapted for generalized geodesic lenses. The final antenna design was validated with commercial electromagnetic simulation software. The antenna combines a modulated geodesic half-MFE lens and a transition to a linear flare, which is needed to preserve the linear polarization in the aperture. The antenna prototype, designed to operate in the K-a-band, was manufactured with computer numerical control (CNC) milling and measured in an anechoic chamber. The design provides continuous beam scanning because of a mechanically actuated feed. Promising beam scanning properties are demonstrated in an angular range of +/- 45 degrees with a scan loss below 3 dB, as well as good frequency stability from 26 to 32 GHz. Since the antenna is fully metallic, its radiation efficiency is high (approximately 90%).