The first order necessary optimality conditions for a weak minimum are derived for optimal control problems with Volterra-type integral equations, considered on a fixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the uniform linear-positive independence of the gradients of active mixed constraints with respect to the control. The conditions obtained generalize the Euler-Lagrange equation (as a stationarity condition) for the Lagrange problem in the classical calculus of variations with ordinary differential equations.