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Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle $C_{4}$ with one chord and $C_{3}^{*}$ be the graph consisting of a triangle with a pendant edge. In this paper we prove that a nontrivial uniquely $C_{4}^{+}$-saturated graph $G$ has girth $3$ or $4$. Further, $G$ has girth $4$ if and only if it is a strongly regular graph with special parameters. 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