The promise of single Cooper-pair quantum circuits based on tunnel junctions for metrology and quantum information applications is severely limited by the influence of offset charges: random, slowly drifting microscopic charges inherent in many solid-state systems. By shunting a small junction with the Josephson kinetic inductance of a series array of large-capacitance tunnel junctions, thereby ensuring that all superconducting islands are connected to the circuit by at least one large junction, we have realized a new superconducting artificial atom that is totally insensitive to offset charges. Yet its energy levels manifest the anharmonic structure associated with single Cooper-pair effects, a useful component for solid-state quantum computation.