There is a very nice book, "Irrational Numbers" by Ivan Niven. Available in paperback from the M.A.A. Evidently he gives a proof in the M.A.A.'s American Mathematical Monthly, volume 46 (1939) pages 469-471. His comment in the notes for chapter 9 of the book has "Proofs of the transcendence of $e$ and $\pi$ are not so difficult as the proof of the more general Theorem 9.1" And his 9.1 is indeed Hermite-LindemannHermite-Lindemann-Weierstrass.  See also http://mathoverflow.net/questions/21367/proof-that-pi-is-transcendental-that-doesnt-use-the-infinitude-of-primes which had a specific emphasis.
There is a very nice book, "Irrational Numbers" by Ivan Niven. Available in paperback from the M.A.A. Evidently he gives a proof in the M.A.A.'s American Mathematical Monthly, volume 46 (1939) pages 469-471. His comment in the notes for chapter 9 of the book has "Proofs of the transcendence of $e$ and $\pi$ are not so difficult as the proof of the more general Theorem 9.1" And his 9.1 is indeed Hermite-Lindemann.  See also http://mathoverflow.net/questions/21367/proof-that-pi-is-transcendental-that-doesnt-use-the-infinitude-of-primes which had a specific emphasis.