  Consider Jacobi random matrix ensembles with the distributions
$$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq
N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N
\left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}}
dx$$ of the eigenvalues on the alcoves $$A:=\{x\in\mathbb R^N| \> -1\leq x_1\le
...\le x_N\leq 1\}.$$ For $(k_1,k_2,k_3)=\kappa\cdot (a,b,1)$ with $a,b>0$
fixed, we derive a central limit theorem for the distributions above for
$\kappa\to\infty$. The drift and the inverse of the limit covariance matrix are
expressed in terms of the zeros of classical Jacobi polynomials. We also
rewrite the CLT in trigonometric form and determine the eigenvalues and
eigenvectors of the limit covariance matrices. These results are related to
corresponding limits for $\beta$-Hermite and $\beta$-Laguerre ensembles for
$\beta\to\infty$ by Dumitriu and Edelman and by Voit.
