Detecting and quantifying chirality is important in fields ranging from analytical and biological chemistry to pharmacology(1) and fundamental physics(2): it can aid drug design and synthesis, contribute to protein structure determination, and help detect parity violation of the weak force. Recent developments employ microwaves(3), femto-second pulses(4), superchiral light(5) or photoionization(6) to determine chirality, yet the most widely used methods remain the traditional methods of measuring circular dichroism and optical rotation. However, these signals are typically very weak against larger time-dependent backgrounds(7). Cavity-enhanced optical methods can be used to amplify weak signals by passing them repeatedly through anoptical cavity, and two-mirror cavities achieving up to 10(5) cavity passes have enabled absorption and birefringence measurements with record sensitivities(8-10). But chiral signals cancel when passing back and forth through a cavity, while the ubiquitous spurious linear birefringence background is enhanced. Even when intracavity optics overcome these problems(11-15), absolute chirality measurements remain difficult and sometimes impossible. Here we use a pulsed-laser bowtie cavity ringdown polarimeter with counter-propagating beams(16,17) to enhance chiral signals by a factor equal to the number of cavity passes (typically > 10(3)); to suppress the effects of linear birefringence by means of a large induced intracavity Faraday rotation; and to effect rapid signal reversals by reversing the Faraday rotation and subtracting signals from the counter-propagating beams. These features allow absolute chiral signal measurements in environments where background subtraction is not feasible: we determine optical rotation from a-pinene vapour in open air, and from maltodextrin and fructose solutions in the evanescent wave produced by total internal reflection at a prism surface. The limits of the present polarimeter, when using a continuous-wave laser locked to a stable, high-finesse cavity, should match the sensitivity of linear birefringence measurements(8) (3 x 10(-13) radians), which is several orders of magnitude more sensitive than current chiral detection limits(7,14,15) and is expected to transform chiral sensing in many fields.