  We present a method for precisely measuring the tunnel splitting in
single-molecule magnets using electron-spin resonance, and use these
measurements to precisely and independently determine the underlying transverse
anisotropy parameter, given a certain class of transitions. By diluting samples
of the SMM Ni$_4$ via co-crystallization in a diamagnetic isostructural
analogue we obtain markedly narrower resonance peaks than are observed in
undiluted samples. Using custom loop-gap resonators we measure the transitions
at several frequencies, allowing a precise determination of the tunnel
splitting. Because the transition under investigation occurs at zero field, and
arises due to a first-order perturbation from the transverse anisotropy, we can
determine the magnitude of this anisotropy independent of any other Hamiltonian
parameters. This method can be applied to other SMMs with tunnel splittings
arising from first-order transverse anisotropy perturbations.
