A formula for evaluating the quadratic normalization integrals of orthogonal Heun functions over the real interval 0 <= x <= 1 is derived using a simple limiting procedure based upon the associated differential equation. The resulting expression gives the value of the normalization integral explicitly in terms of the local power-series solutions about x=0 and x=1 and their derivatives. This provides an extremely efficient alternative to numerical integration for the development of an orthonormal...