Context : We use Planck HFI data combined with ancillary radio data to study the emissivity index of the interstellar dust emission in the frequency range 100–353 GHz , or 3–0.8 mm , in the Galactic plane . We analyse the region l = 20 \degr – 44 \degr and |b| \leq 4 \degr where the free-free emission can be estimated from radio recombination line data . We fit the spectra at each sky pixel with a modified blackbody model and two opacity spectral indices , \beta _ { mm } and \beta _ { FIR } , below and above 353 GHz , respectively . We find that \beta _ { mm } is smaller than \beta _ { FIR } , and we detect a correlation between this low frequency power-law index and the dust optical depth at 353 GHz , \tau _ { 353 } . The opacity spectral index \beta _ { mm } increases from about 1.54 in the more diffuse regions of the Galactic disk , |b| = 3 \degr –4 \degr and \tau _ { 353 } \sim 5 \times 10 ^ { -5 } , to about 1.66 in the densest regions with an optical depth of more than one order of magnitude higher . We associate this correlation with an evolution of the dust emissivity related to the fraction of molecular gas along the line of sight . This translates into \beta _ { mm } \sim 1.54 for a medium that is mostly atomic and \beta _ { mm } \sim 1.66 when the medium is dominated by molecular gas . We find that both the two-level system model and magnetic dipole emission by ferromagnetic particles can explain the results . These results improve our understanding of the physics of interstellar dust and lead towards a complete model of the dust spectrum of the Milky Way from far-infrared to millimetre wavelengths . Aims : Methods : Results : Conclusions : ISM : general – Galaxy : general – radiation mechanisms : general – radio continuum : ISM – submillimeter : ISM