Quantum error correction can allow quantum computers to operate despite the presence of noise and imperfections. A critical component of any error correcting scheme is the mapping of error syndromes onto an ancillary measurement system. However, errors occurring in the ancilla can propagate onto the logical qubit, and irreversibly corrupt the encoded information. Here, we demonstrate a fault-tolerant syndrome measurement scheme that dramatically suppresses forward propagation of ancilla errors. We achieve a fivefold reduction of the logical error probability per measurement, while maintaining the syndrome assignment fidelity. We use the same method to prevent the propagation of thermal ancilla excitations, increasing the logical qubit dephasing time by more than an order of magnitude. Our approach is hardware-efficient, as it uses a single multilevel transmon ancilla and a cavity-encoded logical qubit, whose interaction is engineered in situ using an off-resonant sideband drive. These results demonstrate that hardware-efficient approaches which exploit system-specific error models can yield practical advances towards fault-tolerant quantum computation.