The trace of the vacuum stress tensor is calculated using the Pauli-Villars regularization. A momentum-space calculation up to second order of perturbation theory gives usual expressions for the trace anomalies. The anomalies arise as a contribution of infinitely massive fields (regulators). An important feature of the Pauli-Villars method is that the operations of taking trace and regularization are commutative. This method allows us also to clarify the relation between the massive and massless theories. It is shown that the trace anomaly for a massless field is equal to the negative of the mass-independent term in the expansion ofTvv(x; m2) in powers ofm−2 in the corresponding massive theory.