  In this paper, we propose a Hamiltonian regularization of scalar conservation
laws, which is parametrized by $\ell > 0$ and conserves an $H^1$ energy. We
prove the existence of global weak solutions for this regularization.
Furthermore, we demonstrate that as $\ell$ approaches zero, the unique entropy
solution of the original scalar conservation law is recovered, providing
justification for the regularization. This regularization belongs to a family
of non-diffusive, non-dispersive regularizations that were initially developed
for the shallow-water system and extended later to the Euler system. This paper
represents a validation of this family of regularizations in the scalar case.
