The authors consider a scalar conservation law \(u_ t+\phi(u)_ x=0\) in a single space variable. The initial data is taken to be a finite measure \(\mu\) ; hence the initial condition is stated in the weak form \(\lim_{t\to 0}<\psi(\cdot),u(\cdot,t)>=<\psi(\cdot),d\mu>,\) for all bounded continuous \(\psi\). The motivation for this study comes from the fact that the case where \(\mu\) is the Dirac measure is relevant to the study of the asymptotic behavior of u when \(u(\cdot,0)\) is taken to be an integrable function. The results are different depending on whether u is non-negative, or not, and whether \(\phi\) is odd or convex. In fact, if u is negative somewhere, the problem is not well posed in general, for data which are measures. These differences do not occur if the data is a bounded integrable function. This is an interesting paper having lots of new ideas and it opens up a new area for study.