The development of new frameworks for solid electrolytes that exhibit fast Li-ion diffusion is critical for enabling new energy storage technologies. Here, we present a combined experimental and computational investigation into the ionic conductivity of Li6Y(BO3)3, a new class of solid electrolytes with a pseudo-layered structure. Temperature-dependent impedance spectroscopy shows the pristine material exhibits an ionic conductivity of 2.2×10-3 S/cm around 400°C, while density functional theory calculations point to multiple remarkably low-energy diffusion pathways. Our calculations indicate small energy barriers for lithium interstitials to diffuse along one-dimensional channels oriented in the c-direction, and also for lithium vacancies diffusing within ac planes. This coexistence of diffusion mechanisms indicates that Li6Y(BO3)3 is an extremely versatile host for exploring and understanding mechanisms for lithium-ion conductivity. We find no evidence for reactivity with moisture in the atmosphere and that the material is electrochemically stable when in direct contact with metallic lithium. This robust stability, alongside ionic conductivity that can be manipulated through appropriate aliovalent substitution, make Li6Y(BO3)3 an exceptionally promising new class of solid electrolyte.