In the present study, we have discovered a hitherto undetected fluorescence emission band of rhodamine B in the spectral range of 675 nm to 717 nm for solvents, namely glycerol, ethylene glycol, methanol and butanol. This band could be attributed to H-aggregates (H-dimer) of rhodamine B and it is confirmed by a blue-shifted band at 515 nm in the absorption spectra of rhodamine B in all the solvents. The formation of fluorescent H-aggregates has been further substantiated by the concentration-dependent intensity variation of the absorption and fluorescence peak. Hence, contrary to the previous conceptions that many H-aggregates are naturally non-fluorescent, the pure rhodamine B forms highly fluorescent H-aggregates in glycerol, ethylene glycol, methanol and butanol under ambient conditions. The pH dependent absorption and emission spectra for fixed concentration of rhodamine B in the solvents reveals a red shift in the respective absorption and emission maxima. The aggregation process of the rhodamine B molecule in the solvents has been characterised in terms of its critical concentration which represents the equilibrium phase between the monomers and dimers. A relationship between the critical concentration and the dielectric constant of the solvents has been established and it provides a unique method to predict the critical concentration or the monomer-dimer equilibrium of any other fluorescent dye in a solvent of known dielectric constant. The discovery of the new H-aggregated fluorescence peak has opened a new window for rhodamine B to use in organic solar cells, light-emitting diodes, photographic technology, tunable lasers, and fluorescent depolarization diagnostic devices and as photomedicine. In the monomer-dimer equilibrium state (at critical concentration) of rhodamine B, these applications may be extended up to the new range of radiation uncovered in this work.