The Faddeev equations are solved in momentum space for the trinucleon bound state with the new Tucson-Melbourne π and ρ exchange three-nucleon potentials. The three-nucleon potentials are combined with a variety of realistic two-nucleon potentials. The dependence of the triton binding energy on the πNN cut-off parameter in the three-nucleon potentials is studied and found to be reduced compared to the case with pure π exchange. The ρ exchange parts of the three-nucleon potential yield an overall repulsive effect. When the recommended parameters are employed, the calculated triton binding energy turns out to be very close to its experimental value. Expectation values of various components of the three-nucleon potential are given to illustrate their significance for binding.