The use of oxide fuel cells and other solid-state ionic devices in energy applications is limited by their requirement for elevated operating temperatures, typically above 800 degrees C (ref. 1). Thin-film membranes allow low-temperature operation by reducing the ohmic resistance of the electrolytes(2). However, although proof-of-concept thin-film devices have been demonstrated(3), scaling up remains a significant challenge because large-area membranes less than similar to 100 nm thick are susceptible to mechanical failure. Here, we report that nanoscale yttria-stabilized zirconia membranes with lateral dimensions on the scale of millimetres or centimetres can be made thermomechanically stable by depositing metallic grids on them to function as mechanical supports. We combine such a membrane with a nanostructured dense oxide cathode to make a thin-film solid-oxide fuel cell that can achieve a power density of 155 mW cm(-2) at 510 degrees C. We also report a total power output of more than 20 mW from a single fuel-cell chip. Our large-area membranes could also be relevant to electrochemical energy applications such as gas separation, hydrogen production and permeation membranes.