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
(
η
(
u
)
,
q
(
u
)
)
(\eta (u),q(u))
with
η
(
u
)
\eta (u)
of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in
L
l
o
c
∞
L^\infty _{\mathrm { loc}}
that satisfy the inequality:
η
(
u
)
t
+
q
(
u
)
x
≤
μ
\eta (u)_t+q(u)_x\leq \mu
in the distributional sense for some non-negative Radon measure
μ
\mu
. Furthermore, we extend this result to the class of weak solutions in
L
l
o
c
p
L^p_{\mathrm {loc}}
, based on the asymptotic behavior of the flux function
f
(
u
)
f(u)
and the entropy function
η
(
u
)
\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.