In December 2014, the National Air and Space Museum Archives at the Smithsonian National Air and Space Museum received Arthur C. Clarke papers collection from the Arthur C. Clarke Trust in Sri Lanka. Read about it here, here and here.
I contacted the Smithsonian, and they kindly sent me a photocopy of the Dalai's Lama letter. The letter is short and simple. I am quoting the text, since I'm not sure if I'm allowed to upload the photocopy.
[ Central Tibetan Administration Logo]
THE DALAI LAMA
February 27, 1997
Mr. Arthur C. Clarke, CBE
24 Barnes Place
Dear Mr. Clarke,
Thank you very much for sending me copies of your two books:
1984: SPRING - A Choice of Futures and THE NINE BILLION NAMES OF
GOD. Mr. Jeff Greenwald was kind enough to bring these to me.
Your short story titled "The Nine Billion Names of God" was
Once again, thank you for your thoughtful gesture.
With prayers and good wishes,
Alas (or rather, fortunately) the letter doesn't contain the 9 billionth name after all ;-)