Does a Best Unbiased Estimator for parameter of exponential distribution exist?

Question

For an exponential distribution

$$X \sim \exp(\lambda) = \lambda \ \exp(-\lambda\ x),\ x>0$$

Does there exist an Best Unbiased Estimator (BUE) for $\lambda$ i.e. can it achieve the lower bound for variance given by Cramer-Rao Lower Bound $\lambda^2\, /\,n$?