The DARWIN observatory is a proposed next-generation experiment to search for particle dark matter and for the neutrinoless double beta decay of ¹³⁶Xe. Out of its 50 t total natural xenon inventory, 40 t will be the active target of a time projection chamber which thus contains about 3.6 t of ¹³⁶Xe. Here, we show that its projected half-life sensitivity is 2.4×1027year, using a fiducial volume of 5 t of natural xenon and 10 year of operation with a background rate of less than 0.2 events/(t · year) in the energy region of interest. This sensitivity is based on a detailed Monte Carlo simulation study of the background and event topologies in the large, homogeneous target. DARWIN will be comparable in its science reach to dedicated double beta decay experiments using xenon enriched in ¹³⁶Xe.