  Chemical reactions represent a class of quantum problems that challenge both
the current theoretical understanding and computational capabilities. Reactions
that occur at ultralow temperatures provide an ideal testing ground for quantum
chemistry and scattering theories, as they can be experimentally studied with
unprecedented control, yet display dynamics that are highly complex. Here, we
report the full product state distribution for the reaction 2KRb $\rightarrow$
K$_2$ + Rb$_2$. Ultracold preparation of the reactants grants complete control
over their initial quantum degrees of freedom, while state-resolved, coincident
detection of both products enables the measurement of scattering probabilities
into all 57 allowed rotational state-pairs. Our results show an overall
agreement with a state-counting model based on statistical theory, but also
reveal several deviating state-pairs. In particular, we observe a strong
suppression of population in the state-pair closest to the exoergicity limit,
which we precisely determine to be $9.7711^{+0.0007}_{-0.0005}$ cm$^{-1}$, as a
result of the long-range potential inhibiting the escape of products. The
completeness of our measurements provides a valuable benchmark for quantum
dynamics calculations beyond the current state-of-the-art.
