  We prove existence and uniqueness of Radon measure-valued solutions of the
Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in }
\mathbb{R}\times (0,T) \\ u=u_0\ge 0 &\text{in } \mathbb{R}\times \{0\},
\end{cases} $$ where $u_0$ a positive Radon measure whose singular part is a
finite superposition of Dirac masses, and $\varphi\in C^2([0,\infty))$ is
bounded. The novelty of the paper is the introduction of a compatibility
condition which, combined with standard entropy conditions, guarantees
uniqueness.
