Michael Kremer, a Harvard economics professor with a track record of inventive ideas, and Tom Wilkening, a graduate student at MIT, published a possible solution earlier this year. Instead of flatly banning the export of antiquities, why not ban their sale but allow them to be rented?
The idea has a simple economic justification. Imagine a Malian sculpture, which is currently worth more to the British Museum than to the government of Mali. But it is possible that Mali will be much wealthier in a few decades than it is today and at that point will want the sculpture back. One of the easiest ways to arrange that pattern of possession is for Mali to lease the object to the British Museum for a few decades.
Beyond that smoothly plausible piece of textbook economics, the messy details also point in favor of leasing arrangements. If a poor country protects its antiquities with a blanket export ban, the government has to find cash to dig up, catalog, and store the things. Sensible governments would have other spending priorities, but that then leaves the artifacts in the ground, where they are difficult to protect from smugglers. A leasing arrangement would mean that an impoverished government could invite the Metropolitan Museum or the British Museum to go to all the trouble of excavating, researching, and protecting the treasures in exchange for, say, the right to exhibit them for 25 years.