  Let $X$ be a smooth complex elliptic curve and $G$ a connected reductive
affine algebraic group defined over $\mathbb C$. Let ${\mathcal M}_X(G)$ denote
the moduli space of topologically trivial algebraic $G$--connections on $X$,
that is, pairs of the form $(E_G\, , D)$, where $E_G$ is a topologically
trivial algebraic principal $G$--bundle on $X$, and $D$ is an algebraic
connection on $E_G$. We prove that ${\mathcal M}_X(G)$ does not admit any
nonconstant algebraic function while being biholomorphic to an affine variety.
