A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied.Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh,the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equationand the Hunter--Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions.The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is established.Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation.As the regularisation scale parameter $\ell$ tends to $0$ or $\infty$, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter--Saxton equation respectively, forced by an unknown remaining term.