“The package could not be installed” when updating plugins, themes, and core files on WampServer

Whenever I update anything on my local WordPress development site, whether it be plugins, themes, or the WP 3.7.1 update, I get this error message:

Unpacking the update…

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The package could not be installed.: PCLZIP_ERR_BAD_FORMAT (-10) :
Unable to find End of Central Dir Record signature

Installation Failed

I’m using WampServer 2.4, with Apache 2.0.63, PHP 5.2.11, and MySQL 5.0.88.