We present a novel approach to estimate information carried by experimentally-observed neural spike trains elicited by known stimuli. This approach makes use of an embedding of the observed spike trains into a set of vector spaces, and entropy estimates based on the nearest-neighbor Euclidean distances within these vector spaces (Kozachenko, L.F., and Leonenko, N.N. (1987) Sample estimate of the entropy of a random vector. Problemy Peredachi Informatsii, vol. 23, no. 2, pp. 9-16). Via numerical examples, we show that this approach can be dramatically more efficient than standard bin-based approaches such as the direct method (Strong, S.P., Koberle, R., Ruyter van Steveninck, R.R. de, and Bialek, W. (1998) Entropy and information in neural spike trains. Phys. Rev. Lett. 80, 197-200) for amounts of data typically available from laboratory experiments.