Let f be a function analytic in a multiply connected region, D, containing the simple closed contour C. Approximations to the contour integral of f around C by weighted sums of function values are considered. For any fixed choice of distinct evaluation points, the weights can be chosen so that the quadrature is exact for elements of a linear space of rational functions whose poles are known. A convergence theorem involving a trapezoidal distribution of points for the quadrature, around C, of functions analytic in D is established. Also, an analogue of the Peano remainder theorem can be deduced which extends previous work of Lyness and Delves [6].