A devolatilizer apparatus and method for devolatilization of viscous polymer liquids yields polymer products with very low levels of residual volatile components. A stream of liquid polymer is dropped through a first vacuum chamber, thereby removing a portion of the volatile components. The stream is collected at the bottom of the first chamber, and is re-circulated to a manifold and liquid distributor assembly in a second chamber in the same vessel. The stream flows by gravity through the second chamber, which is maintained at a higher level of vacuum than the first chamber. Improved devolatilization is accomplished by exposing the polymer liquid to multiple stages of vacuum in a single vessel. The first chamber may be located generally above the second chamber, but in the same vessel, with the stream falling through the second chamber a first time before re-circulation to the manifold.