  We show that several families of classical orthogonal polynomials on the real
line are also orthogonal on the interior of an ellipse in the complex plane,
subject to a weighted planar Lebesgue measure. In particular these include
Gegenbauer polynomials $C_n^{(1+\alpha)}(z)$ for $\alpha>-1$ containing the
Legendre polynomials $P_n(z)$, and the subset
$P_n^{(\alpha+\frac12,\pm\frac12)}(z)$ of the Jacobi polynomials. These
polynomials provide an orthonormal basis and the corresponding weighted Bergman
space forms a complete metric space. This leads to a certain family of Selberg
integrals in the complex plane. We recover the known orthogonality of Chebyshev
polynomials of first up to fourth kind. The limit $\alpha\to\infty$ leads back
to the known Hermite polynomials orthogonal in the entire complex plane. When
the ellipse degenerates to a circle we obtain the weight function and monomials
known from the determinantal point process of the ensemble of truncated unitary
random matrices.
