The best known bounds seem to be due to Nicely [``A new error analysis for Brun's constant,'' Virginia J. Sci. \textbf{52} (2001), no. 1, 45–55), who showed that Brun's constant is $$ 1.9021605823 \pm 0.0000000008 $$ To do so he computed all prime twins up to $3 \times 10^{15}$. Since the known upper bound for prime twins differs from the expected number by a factor of 4 (better bounds are known, but making them explicit would probably be extremely difficult), it is likely that the only possible way to improve on this would be to extend the range of computation.
Edit. The article by Nicely can be obtained here: http://www.trnicely.net/twins/twins4.html . As nautilus already remarked, these computations do not give a proven bound, but a heuristic estimate, and the computations by Klyve are superior anyway. I took my information from MathSciNet, which is not always correct.
