Weir Farm National Historic Site Amendment Act - Amends the Weir Farm National Historic Site Establishment Act of 1990 to: (1) require that property acquired for the development of visitor and administrative facilities for the Weir Farm National Historic Site in Connecticut be within Fairfield county; (2) continue to require minimum development of the property and prohibit anything from either preventing the acquisition of property for the development of such facilities that, prior to the acquisition, was developed in a manner inconsistent with such requirement, or requiring remediation of such previously developed property to reflect the natural character of the property comprising the Site; and (3) require, prior to and as a prerequisite to any development of visitor and administrative facilities on the acquired property, one or more agreements with the chief elected official of the unit of local government having jurisdiction over such property.