We present a new " cover and decomposition " attack on the elliptic curve discrete logarithm problem, that combines Weil descent and decomposition-based index calculus into a single discrete logarithm algorithm. This attack applies, at least theoretically, to all composite degree extension fields, and is particularly well-suited for curves defined over Fp6. We give a real-size example 3 of discrete logarithm computations on a curve over a 156-bit degree 6 extension field, which would not have been practically attackable using previously known algorithms. A shorter version of this work was presented at the EUROCRYPT 2012 conference.