  A critical challenge in developing scalable error-corrected quantum systems
is the accumulation of errors while performing operations and measurements. One
promising approach is to design a system where errors can be detected and
converted into erasures. Such a system utilizing erasure qubits are known to
have relaxed requirements for quantum error correction. A recent proposal aims
to do this using a dual-rail encoding with superconducting cavities. However,
experimental characterization and demonstration of a dual-rail cavity qubit has
not yet been realized. In this work, we implement such a dual-rail cavity
qubit; we demonstrate a projective logical measurement with integrated erasure
detection and use it to measure dual-rail qubit idling errors. We measure
logical state preparation and measurement errors at the $0.01\%$-level and
detect over $99\%$ of cavity decay events as erasures. We use the precision of
this new measurement protocol to distinguish different types of errors in this
system, finding that while decay errors occur with probability $\sim 0.2\%$ per
microsecond, phase errors occur 6 times less frequently and bit flips occur at
least 140 times less frequently. These findings represent the first
confirmation of the expected error hierarchy necessary to concatenate dual-rail
erasure qubits into a highly efficient erasure code.
