The need to evaluate Gauss quadrature rules arises in many applications in science and engineering. It often is important to be able to estimate the quadrature error when applying an ℓ-point Gauss rule, Gℓ(f), where f is an integrand of interest. Such an estimate often is furnished by applying another quadrature rule, Qk(f), with k>ℓ nodes, and using the difference Qk(f)−Gℓ(f) or its magnitude as an estimate for the quadrature error in Gℓ(f) or its magnitude. The classical approach to estimate the error in Gℓ(f) is to let Qk(f), with k=2ℓ+1, be the Gauss-Kronrod quadrature rule associated with Gℓ(f). However, it is well known that the Gauss-Kronrod rule associated with a Gauss rule Gℓ(f) might not exist for certain measures that determine the Gauss rule and for certain numbers of nodes. This prompted M. M. Spalević 1 to develop generalized averaged Gauss rules, Gˆ2ℓ+1, with 2ℓ+1 nodes for estimating the error in Gℓ(f). Similarly as for (2ℓ+1)-node Gauss-Kronrod rules, ℓ nodes of the rule Gˆ2ℓ+1 agree with the nodes of Gℓ. However, generalized averaged Gauss rules are not internal for some measures. They therefore may not be applicable when the integrand only is defined on the convex hull of the support of the measure. This paper describes a new kind of quadrature rules that may be internal also when generalized averaged quadrature rules are not. The construction of the new quadrature rules is based on theory developed by Peherstorfer 2. Their application is particularly attractive when the rule Gˆ2ℓ+1 is not internal, the integrand cannot be evaluated at all its nodes, and the integrand is inexpensive to evaluate at the quadrature points. Computed examples that illustrate the performance of the new quadrature rules introduced in this paper are presented.