Recently, a stability theorem for the Feynman integral as a bounded linear operator on$ L_2$($R^{d}$ /) with respect to measures whose positive and negative variations are in the generalized Kato class was proved. We study a stability theorem for the Feynman integral with respect to measures whose positive variations are in the class of $sigma$-finite smooth measures and negative variations are in the generalized Kato class. This extends the recent result in the sense that the class of $sigma$-finite smooth measures properly contains the generalized Kato class.