For a real interval polynomial F, we provide a rigorous method for deciding whether there exists a polynomial in F that has a multiple zero in a prescribed interval in R. We show that it is sufficient to examine a finite number of edge polynomials in F. An edge polynomial is a real interval polynomial such that the number of coefficients that are intervals is one. The decision method uses the property that a univariate polynomial is of degree one with respect to each coefficient regarded as a variable. Using this method, we can completely determine the set of real numbers each of which is a multiple zero of some polynomial in F.