We provide conditions on an exchange economy with asymmetric information that guarantee that when the economy is replicated sufficiently often, there will be an allocation which is incentive compatible, individually rational, and nearly efficient. The main theorem covers both the case in which aggregate uncertainty remains when the economy is replicated and the case in which replication eliminates aggregate uncertainty. In addition, we demonstrate how our theorem does or does not apply to standard asymmetric information problems such as the buyer's bid double auction problem, Akerlof's lemons problem, and insurance with asymmetric information.