We present a novel class of compact, robust, and potentially cost-effective chaotic cavities for trace-gas sensing applications. The cavity consists of a single, closed, reflecting surface and can support up to 15.5 meter of optical path length with a cavity volume of only 68 cm3. The design is based on quasi-chaotic ray dynamics in a 3-dimensional (3-D) deformed sphere. Within the cavity, light traces out a quasi-stable trajectory that has been improved to achieve long optical path length and little beam overlap. The results show that the proposed chaotic cavity effectively refocuses the beam as the light bounces off the cavity wall. We modeled, fabricated, and tested a prototype cavity. The chaotic cavity can successfully achieve a 40-pass trajectory with a pulsed AlGaInP diode laser emitting at 661 nm. The experimental results agree well with calculations.