We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair $(\eta(u),q(u))$ with $\eta(u)$ of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in $L_{loc}^\infty$ that satisfy the inequality: $\partial_t\eta(u)+\partial_x q(u)\leq \mu$ in the distributional sense for some non-negative Radon measure $\mu$. Furthermore, we extend this result to the class of weak solutions in $L_{loc}^p$, based on the asymptotic behavior of the flux function $f(u)$ and the entropy function $\eta(u)$ at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness. This is a joint work with Gui-Qiang G. Chen.