Let X be a Kähler manifold and ∆ be a R-divisor with simple normal crossing support and coefficients between 1/2 and 1. Assuming that K X + ∆ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X \ Supp(∆) having mixed Poincaré and cone singularities according to the coefficients of ∆. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair (X, ∆).