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EDIT. There are very many published proofs. For example, Colin de Verdiere (Forum Math., 1 (1989) 395-402), or Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256.

Thurston's original proof is elementary: he gives an algorithm for finding the (essentially unique) circle packing from a triangulation. He never published this properly; his "Geometry and Topology of 3-manifolds" exists as a preprint, and the proof there are sketchy. Then Colin de Verdiere (Forum Math., 1 (1989) 395-402) proved accurately convergence of Thurston's algorithm. Another proof of convergence and implementation on computer is in the paper of Collins and Stephenson mentioned above. Then there were many generalizations and versions of this theorem and algorithm, too many to list all of them here.

EDIT. There are very many published proofs. For example, Colin de Verdière (Forum Math., 1 (1989) 395-402), or Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256.

Thurston's original proof is elementary: he gives an algorithm for finding the (essentially unique) circle packing from a triangulation. He never published this properly; his "Geometry and Topology of 3-manifolds" exists as a preprint, and the proof there are sketchy. Then Colin de Verdière (Forum Math., 1 (1989) 395-402) proved accurately convergence of Thurston's algorithm. Another proof of convergence and implementation on computer is in the paper of Collins and Stephenson mentioned above. Then there were many generalizations and versions of this theorem and algorithm, too many to list all of them here.

There are very many published proofs. For example, B. Rodin and D. Sullivan, THE CONVERGENCE OF CIRCLE PACKINGS TO THE RIEMANN MAPPING, J. DIFFERENTIAL GEOMETRY 26 (1987) 349-360. Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256, and many generalizations. Just look at the references on the Rodin and Sullivan's paper on Mathscinet or elsewhere.

EDIT. Thurston's original proof is elementary: he gives an algorithm for finding the (essentially unique) circle packing from a triangulation. He never published this properly; his "Geometry and Topology of 3-manifolds" exists as a preprint, and the proof there are sketchy. The first published complete proof (I believe) is of Rodin and Sullivan. Then Colin de Verdiere (Forum Math., 1 (1989) 395-402) proved accurately convergence of Thurston's algorithm. Another proof of convergence and implementation on computer is in the paper of Collins and Stephenson mentioned above. Then there were many generalizations and versions of this theorem and algorithm, too many to list all of them here.

EDIT. There are very many published proofs. For example, Colin de Verdiere (Forum Math., 1 (1989) 395-402), or Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256.

Thurston's original proof is elementary: he gives an algorithm for finding the (essentially unique) circle packing from a triangulation. He never published this properly; his "Geometry and Topology of 3-manifolds" exists as a preprint, and the proof there are sketchy. Then Colin de Verdiere (Forum Math., 1 (1989) 395-402) proved accurately convergence of Thurston's algorithm. Another proof of convergence and implementation on computer is in the paper of Collins and Stephenson mentioned above. Then there were many generalizations and versions of this theorem and algorithm, too many to list all of them here.

There are very many published proofs. For example, B. Rodin and D. Sullivan, THE CONVERGENCE OF CIRCLE PACKINGS TO THE RIEMANN MAPPING, J. DIFFERENTIAL GEOMETRY 26 (1987) 349-360. Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256, and many generalizations. Just look at the references on the Rodin and Sullivan's paper on Mathscinet or elsewhere.

There are very many published proofs. For example, B. Rodin and D. Sullivan, THE CONVERGENCE OF CIRCLE PACKINGS TO THE RIEMANN MAPPING, J. DIFFERENTIAL GEOMETRY 26 (1987) 349-360. Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256, and many generalizations. Just look at the references on the Rodin and Sullivan's paper on Mathscinet or elsewhere.

EDIT. Thurston's original proof is elementary: he gives an algorithm for finding the (essentially unique) circle packing from a triangulation. He never published this properly; his "Geometry and Topology of 3-manifolds" exists as a preprint, and the proof there are sketchy. The first published complete proof (I believe) is of Rodin and Sullivan. Then Colin de Verdiere (Forum Math., 1 (1989) 395-402) proved accurately convergence of Thurston's algorithm. Another proof of convergence and implementation on computer is in the paper of Collins and Stephenson mentioned above. Then there were many generalizations and versions of this theorem and algorithm, too many to list all of them here.