Absorption, fluorescence emission and excitation spectra of benzothiazole cyanine dyes--thiazole orange (TO) and 7-methyl-6-(3-methyl-2,3-dihydro-1,3-benzothiazol-2-ylidenmethyl) [1,3] dioxolo [4',5':4,5] benzo [d] [1,3] thiazolium methylmethosulfate (Cyan 13)--were investigated over a wide concentration range. The dyes form aggregates with a 'sandwich'-like structure in water solution. At low dye to DNA concentrations ratios, Cyan 13 and TO monomers appear to interact with the DNA. On increasing the dye to DNA concentrations ratio, free dye molecules aggregate with the DNA-bound ones. The spectra of the free dye aggregates and the aggregates formed on the DNA, are characterized by an anomalously large (more than 100 nm) Stokes shift. This suggests, that the pi-electron systems of the aggregates undergo substantial changes in excited state, compared to those of the monomers. The formation of aggregates consisting of the free and DNA-bound dye molecules can be explained using the half-intercalation model of the interaction of the cyanine dye monomers with the DNA.