Summary: We show that the symmetrized bidisc \({\mathbb G}_2= \{(\lambda_1+\lambda_2, \lambda_1\lambda_2):|\lambda_1| ,|\lambda_2| <1\}\subset{\mathbb C}^2\) cannot be exhausted by domains biholomorphic to convex domains improving the result of C. Costara in Bull. Lond. Math. Soc. 36, No. 5, 656–662 (2004; Zbl 1065.32006).