We use neutron scattering to show that the low-temperature, short-range ordered spin configuration in the geometrically frustrated magnet SrCr_9pGa_{12 − 9p}O₁₉ (p = 0.92(5)) is composed of small groups of spins whose dipole moments cancel. The local magnetic fluctuation spectrum, χ^′′(ω), vanishes approximately in proportion to ω for ℏω < 0.5 meV, distinguishing this magnet from conventional spin glasses which display a featureless continuum of excited states. We argue that this behavior results from the absence of local, low-energy excitations in the zero-spin clusters from which the frozen spin configuration is composed.