The structural properties of LaCu6−xAux are studied using neutron diffraction, x-ray diffraction, and heat capacity measurements. The continuous orthorhombic-monoclinic structural phase transition in LaCu6 is suppressed linearly with Au substitution until a complete suppression of the structural phase transition occurs at the critical composition xc=0.3 . Heat capacity measurements at low temperatures indicate residual structural instability at xc . The instability is ferroelastic in nature, with density functional theory calculations showing negligible coupling to electronic states near the Fermi level. The data and calculations presented here are consistent with the zero temperature termination of a continuous structural phase transition suggesting that the LaCu6−xAux series hosts an elastic quantum critical point.