  We describe and implement a method to restore the state of a single qubit, in
principle perfectly, after it has partially collapsed. The method resembles the
classical Hahn spin-echo, but works on a wider class of relaxation processes,
in which the quantum state partially leaves the computational Hilbert space. It
is not guaranteed to work every time, but successful outcomes are heralded. We
demonstrate using a single trapped ion better performance from this recovery
method than can be obtained employing projection and post-selection alone. The
demonstration features a novel qubit implementation that permits both partial
collapse and coherent manipulations with high fidelity.
