An inequality relating the parameter ${\lambda }_{+}$ of $K_{l3\mathrm{}}$ decays to the spectral function of the isovector electromagnetic current is derived, assuming only Weinberg's first spectral-function sum rule. It is shown that a large value for ${\lambda }_{+}$ (e.g., ${\lambda }_{+}=0.045\mathrm{}$) is not compatible with $\rho$ saturation of the isovector spectral function. Since the principal tests of the sum rule have relied on the additional assumption that such resonance saturations are possible, some doubt is cast on the validity, or at least on the previous applications, of the sum rule. Implications for the Weinberg mass relations, $c$-number Schwinger terms, and the $e^{+}{e}^{-}$- annihilation cross section are discussed.

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