In the search for high-performance thermoelectrics, materials such as clathrates have drawn attention due to having both glasslike low phonon thermal conductivity and crystal-like high electrical conductivity. ${\mathrm{Ba}}_{8}{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ (BGG) has a loosely bound guest Ba atom trapped inside rigid Ga-Ge cage structures. Avoided crossings between acoustic phonons and the flat guest atom branches have been proposed to be the source of the low lattice thermal conductivity of BGG. Ga-Ge site disorder with Ga and Ge exchanging places in different unit cells has also been reported. We used time-of-flight neutron scattering to measure the complete phonon spectrum in a large single crystal of BGG and compared these results with predictions of density functional theory to elucidate the effect of the disorder on heat-carrying phonons. Experimental results agreed much better with the calculation assuming the disorder than with the calculation assuming the ordered configuration. Although the atomic masses of Ga and Ge are nearly identical, we found that disorder strongly reduces phonon group velocities, which significantly reduces thermal conductivity. Our work points to a path towards optimizing thermoelectrics.