  A long-standing conjecture in quantum field theory due to Broadhurst and
Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit
rational multiple of the odd values of the Riemann zeta function. In this paper
we prove this conjecture by constructing a certain family of single-valued
multiple polylogarithms. The zig-zag graphs therefore provide the only infinite
family of primitive graphs in $\phi^4_4$ theory (in fact, in any renormalisable
quantum field theory in four dimensions) whose amplitudes are now known.
