Ultrasound-based sensing of muscle deformation, known as sonomyography, has shown promise for accurately classifying the intended hand grasps of individuals with upper limb loss in offline settings. Building upon this previous work, we present the first demonstration of real-time prosthetic hand control using sonomyography to perform functional tasks. An individual with congenital bilateral limb absence was fitted with sockets containing a low-profile ultrasound transducer placed over forearm muscle tissue in the residual limbs. A classifier was trained using linear discriminant analysis to recognize ultrasound images of muscle contractions for three discrete hand configurations (rest, tripod grasp, index finger point) under a variety of arm positions designed to cover the reachable workspace. A prosthetic hand mounted to the socket was then controlled using this classifier. Using this real-time sonomyographic control, the participant was able to complete three functional tasks that required selecting different hand grasps in order to grasp and move one-inch wooden blocks over a broad range of arm positions. Additionally, these tests were successfully repeated without retraining the classifier across 3 hours of prosthesis use and following simulated donning and doffing of the socket. This study supports the feasibility of using sonomyography to control upper limb prostheses in real-world applications.