This paper considers compact embedded surfaces \(\Sigma\) of constant mean curvature and with \(\partial{\Sigma}\) a planar Jordan curve. In this case, \(\partial{\Sigma}\) bounds a domain \(D\) in \(\{x_3=0\}\). It is shown that, if for some neighborhood \(N\) of \(\partial{\Sigma}\) in \({\mathbb R}^3\) one has that \(N\cap \Sigma\) is a non-negative graph over a neighborhood of \(\partial{\Sigma}\) in \(\overline{D}\), then \(\text{ int } \Sigma\) is a positive graph over \(D\). The author uses the terminology \(\Sigma\) is ``locally a graph over'' \(D\) to describe the hypotheses stated above. In particular, he means ``over'' to convey non-negativity (and, sometimes, non-positivity). Technically, he sometimes assumes strict positivity, but this is not really necessary for the proof. The property of being locally a non-negative graph around \(\partial{\Sigma}\) is used in conjunction with an integral balancing formula to show that \(N\cap \Sigma\) has negative mean curvature with respect to its downward pointing normal. (This reasoning was essentially contained in the proof of Theorem~1 in the paper of \textit{R. Sa Earp, F. Brito, W. H. Meeks III} and \textit{H. Rosenberg}, Indiana Univ. Math. J. 40, 333-343 (1991; Zbl 0759.53003)].) It then follows from the Hopf boundary point lemma that \(\Sigma\) is transverse to \(\{x_3=0\}\) along \(\partial{\Sigma}\). The author then quotes Theorem~2 from the paper of Brito, Earp, et al. mentioned above to assert that \(\Sigma \cap D=\phi\). The proof is then completed with an application of the Alexandrov reflection procedure (which one should learn from the papers of Alexandrov, Wente, and Serrin before reading this paper). This particular application is somewhat similar to those in the E-B-M-R paper and in the recent paper [\textit{A. Ros} and \textit{H. Rosenberg}, J. Differ. Geom. 44, 807-817 (1996; Zbl 0883.53009)].