Pop proved that a smooth curve C over an ample held K with C(K) not equal aleph has vertical bar K vertical bar many rational points. We strengthen this result by showing that there are vertical bar K vertical bar many rational points that do not lie in a given proper subfield, even after applying a rational map. This has several consequences. For example, we gain insight into the Structure of existentially definable (i.e. diophantine) subsets of ample fields. (C) 2009 Elsevier Inc. All rights reserved.