The construction of a perpetual motion machine has remained an unfulfilled objective in the mechanical and physical sciences community. Despite numerous attempts over the centuries, none has achieved success. At present, such machines are considered impossible as they appear to contradict the principle of energy conservation, a fundamental pillar for modern physics. This paper presents a theoretical solution to this ancient problem, relying on a previously unknown exception to the principle of energy conservation: Specifically, energy is conservatively transferred from a primary to a secondary mechanical motion, while the primary motion appears to recover its lost energy through angular momentum conservation. This process leads to energy being produced out of nothing. The proposed ideal machine is established within the framework of classical Newtonian mechanics, without considering any relativistic effects. Its function is proven mathematically, and expressions are derived to determine efficiency. The results reveal that a non-trivial amount of energy could be produced in this manner. In this context, this study extends beyond intellectual curiosity and suggests a new principle for developing energy-generating machines with valuable and practical implications.