In this paper, we prove that the representation $\rho$ from $G_{\mathbb{Q}}$ in GL${}_2\left(ℂ\right)$ with image $A_5$ in PGL${}_2\left(A_5\right)$ corresponding to the example $16$ in [B-K] is modular. This representation has conductor $5203$ and determinant ${\chi }_{-43}$; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].