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Context. Carbon monoxide (CO) is a poor tracer of H₂in the diffuse interstellar medium (ISM), where most of the carbon is not incorporated into CO molecules, unlike the situation at higher extinctions. Aims. We present a novel, indirect method for constraining H₂column densities (N_H2) without employing CO observations. We show that previously recognized nonlinearities in the relation between the extinction,A_V(H₂), derived from dust emission and the H Icolumn density (N_H I) are due to the presence of molecular gas. Methods. We employed archival (N_H2) data, obtained from the UV spectra of stars, and calculatedA_V(H₂) toward these sight lines using 3D extinction maps. The following relation fits the data: logN_H2= 1.38742 (logA_V(H₂)⁾³− 0.05359 (logA_V(H₂)⁾²+ 0.25722 logA_V(H₂) + 20.67191. This relation is useful for constrainingN_H2in the diffuse ISM as it requires onlyN_H Iand dust extinction data, which are both easily accessible. In 95% of the cases, the estimates produced by the fitted equation have deviations of less than a factor of 3.5. We constructed aN_H2map of our Galaxy and compared it to the CO integrated intensity (W_CO) distribution. Results. We find that the average ratio (X_CO) betweenN_H2andW_COis approximately equal to 2 × 10²⁰cm⁻²(K km s⁻¹)⁻¹, consistent with previous estimates. However, we find that theX_COfactor varies by orders of magnitude on arcminute scales between the outer and the central portions of molecular clouds. For regions withN_H2≳ 10²⁰cm⁻², we estimate that the average H₂fractional abundance,f_H2= 2N_H2/(2N_H2+N_H I), is 0.25. Multiple (distinct) largely atomic clouds are likely found along high-extinction sightlines (A_V≥ 1 mag), hence limitingf_H2in these directions. Conclusions. More than 50% of the lines of sight withN_H2≥ 10²⁰cm⁻²are untraceable by CO with aJ= 1−0 sensitivity limitW_CO= 1 K km s⁻¹.