Antenna-pattern measurements obtained from a double-metal supra-terahertz-frequency (supra-THz) quantum cascade laser (QCL) are presented. The QCL is mounted within a mechanically micro-machined waveguide cavity containing dual diagonal feedhorns. Operating in continuous-wave mode at 3.5 THz, and at an ambient temperature of ~60 K, QCL emission has been directed via the feedhorns to a supra-THz detector mounted on a multi-axis linear scanner. Comparison of simulated and measured far-field antenna patterns shows an excellent degree of correlation between beamwidth (full-width-half-maximum) and sidelobe content and a very substantial improvement when compared with unmounted devices. Additionally, a single output has been used to successfully illuminate and demonstrate an optical breadboard arrangement associated with a future supra-THz Earth observation space-borne payload. Our novel device has therefore provided a valuable demonstration of the effectiveness of supra-THz diagonal feedhorns and QCL devices for future space-borne ultra-high-frequency Earth-observing heterodyne radiometers.