For a function \(f\in L^{\infty}(T^ 2)\), T the unit circle, \textit{S.-Y. A. Chang} has proved [Ann. Math., II. Ser. 109, 613-620 (1979; Zbl 0401.28004)] that the Poisson integral \(\Lambda\) is a bounded operator from \(L^ 2(T^ 2)\) to \(L^ 2(d\mu_ f)\), where \[ d\mu_ f= | \nabla_ 1\nabla_ 2 \Lambda f(z_ 1,z_ 2)|^ 2 \log (1/| z_ 1|) \log (1/| z_ 2|) dV(z_ 1) dV(z_ 2) \] and where dV is area measure on the disk. The present author proves that if \(f\in C(T^ 2)\), then the above operator \(\Lambda\) is compact. An immediate consequence is an extension of a one variable result of \textit{S. C. Power} [Bull. Lond. Math. Soc. 12, 207-210 (1980; Zbl 0438.47033)].