Anti-B-B Mixing Constrains Topcolor-Assisted Technicolor

We argue that extended technicolor augmented with topcolor requires that all mixing between the third and the first two quark generations resides in the mixing matrix of left-handed down quarks. Then, the anti-B_d--B_d mixing that occurs in topcolor models constrains the coloron and Z' boson masses to be greater than about 5 TeV. This implies fine tuning of the topcolor couplings to better than 1%.