  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 equation and 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.
