Abstract: The classical Gabriel-Ulmer duality asserts a dual adjoint equivalence between finitely complete categories, and a full sub-2-category of the complete, filtered-cocomplete categories which are known as locally finitely presentable (LFP). The definition of LFP category involves exactness conditions'' asserting compatibility between limits and filtered colimits, together with a different sort of condition which amounts to admitting enough structure-preserving functors to Set; removing this last condition yields the continuous locally presentable (CLP) categories in the sense of Johnstone-Joyal. We prove an analog of Gabriel-Ulmer duality for all CLP categories, by replacing the dualizing category Set with the category CPUMet of complete partial ultrametric spaces. As with Gabriel-Ulmer duality, this result has a logical interpretation, as a strong conceptual completeness theorem for the lex fragment'' of a continuous first-order logic for partial ultrametric structures.