We prove a new explicit inequality for the non-dimensional flow force constant, significantly improving the Benjamin and Lighthill conjecture about irrotational steady water waves. As a corollary, we prove a bound for the wave amplitude in terms of the Bernoulli constant. We show that the amplitude decays as $r^{-2}$ when $r \to +\infty$, where $r$ is the non-dimensional Bernoulli constant. We explain that the latter limit corresponds to deep water waves and the bound for the amplitude is sharp. In terms of physical parameters the result states that the amplitude $a$ of an arbitrary Stokes wave is bounded by $C m^2 g/Q^2$, where $m$ is the relative mass flux, $g$ is the gravitational constant, $Q$ is the total head and $C$ is an absolute constant given explicitly. In particular, this implies that $a < C c^2 g^{-1}$, where $c$ is the wave speed. The latter inequality is valid for all Stokes waves, irrespective of wavelength or amplitude, including extreme waves.