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X-ray imaging, based on ionizing radiation, can be used to determine bone age by examining distal growth plate fusion in the ulna and radius bones. Legal age determination approaches based on ultrasound signals exist but are unsuitable to reliably determine bone age. We present a low-cost, mobile system that uses one-dimensional ultrasound radio frequency signals to obtain a robust binary classifier enabling the determination of bone age from data of girls and women aged 9 to 24 years. These data were acquired as part of a clinical study conducted with 148 subjects. Our system detects the presence or absence of the epiphyseal plate by moving ultrasound array transducers along the forearm, measuring reflection and transmission signals. Even though classical digital signal processing methods did not achieve a robust classifier, we achieved an F1 score of approximately 87% for binary classification of completed bone growth with machine learning approaches, such as the gradient boosting machine method CatBoost. We demonstrate that our ultrasound system can classify the fusion of the distal growth plate of the radius bone and the completion of bone growth with high accuracy. We propose a non-ionizing alternative to established X-ray imaging methods for this purpose.