  The environment-induced dissipation of an open system, once thought as a
nuisance, can actually lead to emergence of many intriguing phenomena that are
absent in an isolated system. Among these, Weyl exceptional rings (WER),
extended from point-like singularities, are particularly interesting.
Theoretically, a WER was predicted to carry a topological charge with a nonzero
Chern number, but it has not been measured so far. We here investigate this
topology in a circuit, where the WER is synthesized with a superconducting
qubit controllably coupled to a decaying resonator. The high flexibility of the
system enables us to characterize its eigenvectors on different manifolds of
parameter space. We extract both the quantized Berry phase and Chern number
from these eigenvectors. Furthermore, we demonstrate a topological transition
triggered by shrinking the size of the manifold$-$a unique feature of the WER.
