We gave up sovereignty of the southern US border with the La Paz agreement, and the border has been administered by a joint commission in Mexico City for about twenty years:
http://www.nmsu.edu/~bec/BEC/Readings/10.USMBHC-TheBorderAtAGlance.pdf
The La Paz Agreement deals specifically with environmental issues of concern to the EPA and to its sister agency within the Mexican government. The Agreement has nothing to do with—nor does it infringe upon the sovereignty of—the U.S. border with Mexico.