As it is mentioned above, you can use optical absorbance measurement. The onset of the absorbance gives you so-called optical energy gap. The other simple and popular way is to use electrochemical methods such as cyclic voltammetry or steady-state voltammetry. There you can obtain not the energy gap only, but it is possible to estimate the conduction band and valence band (or HOMO and LUMO) positions. In that case it is necessary to select appropriate electrolyte. Very often is used acetonitrile with tetrabutylammonium hexafluorophosphate (to increase the conductivity).

If you a have a spectrophotometer you can plot the absorption spectra as a function of the energy photons. The intercept on the ex axix will give you the value of the energy bandgap.

As it is mentioned above, you can use optical absorbance measurement. The onset of the absorbance gives you so-called optical energy gap. The other simple and popular way is to use electrochemical methods such as cyclic voltammetry or steady-state voltammetry. There you can obtain not the energy gap only, but it is possible to estimate the conduction band and valence band (or HOMO and LUMO) positions. In that case it is necessary to select appropriate electrolyte. Very often is used acetonitrile with tetrabutylammonium hexafluorophosphate (to increase the conductivity).

plot a graph between alpha^1/2 Vs eV, tangent on the curve which cut it at x axis will be the band gap

You may also use photoelectron spectroscopy (quite popular especially for the energy gap in superconductors, for reference, see e.g. Appl. Phys. Lett. 63 (1993) 2911-2913) and, in addition to what Martin has mentioned above, you may combine electrochemistry with illumination (photo-electrochemistry) and may also get the band gap from such kind of experiments (see e.g. J. Elecrroanal. Chem., 210 (1986) 213-227 , 280 (1990) 91-103 or 284 (1990) 385-401 for more information on this).