Tiny "gnat robots," weighing just a few milligrams, were first conjectured in the 1980s. How to stabilize one if it were to hover like a small insect has not been answered. Challenges include the requirement that sensors be both low mass and high bandwidth and that silicon-micromachined rate gyroscopes are too heavy. The smallest robot to perform controlled hovering uses a sensor suite weighing hundreds of milligrams. Here, we demonstrate that an accelerometer represents perhaps the most direct way to stabilize flight while satisfying the extreme size, speed, weight, and power constraints of a flying robot even as it scales down to just a few milligrams. As aircraft scale reduces, scaling physics dictates that the ratio of aerodynamic drag to mass increases. This results in reduced noise in an accelerometer's airspeed measurement. We show through simulation and experiment on a 30-gram robot that a 2-milligram off-the-shelf accelerometer is able in principle to stabilize a 10-milligram robot despite high noise in the sensor itself. Inspired by wind-vision sensory fusion in the flight controller of the fruit fly Drosophila melanogaster, we then added a tiny camera and efficient, fly-inspired autocorrelation-based visual processing to allow the robot to estimate and reject wind as well as control its attitude and flight velocity using a Kalman filter. Our biology-inspired approach, validated on a small flying helicopter, has a wind gust response comparable to the fruit fly and is small and efficient enough for a 10-milligram flying vehicle (weighing less than a grain of rice).