The loop space $L\bP_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps $S^1 \to \bP_1$ is an infinite dimensional complex manifold. The loop group $L\pgl$ acts on $L\bP_1$. We prove that the group of $L\pgl$-invariant holomorphic line bundles on $L\bP_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of any such line bundle is finite dimensional, and compute the dimension for a generic bundle.