We prove that if BPP≠EXP, then every problem in BPP can be solved deterministically in subexponential time on almost every input (on every sampleable ensemble for infinitely many input sizes). This is the first derandomization result for BPP based on uniform, noncryptographic hardness assumptions. It implies the following gap in the deterministic average-case complexities of problems in BPP: either these complexities are always sub-exponential or they contain arbitrarily large exponential functions. We use a construction of a small “pseudorandom” set of strings from a “hard function” in EXP which is identical to that used in the analogous nonuniform results in L. Babai et al. (Comput. Complexity3 (1993), 307–318) and N. Nisan and A. Wigderson (J. Comput. System Sci.49 (1994), 149–167). However, previous proofs of correctness assume the “hard function” is not in P/poly. They give a non constructive argument that a circuit distinguishing the pseudorandom strings from truly random strings implies that a similarly sized circuit exists computing the “hard function.” Our main technical contribution is to show that, if the “hard function” has certain properties, then this argument can be made constructive.