I'm not sure I understand what the problem is. If you know the molecular weight, you weigh out the sample, dissolve in a known volume of solvent, take an absorption spectrum. You'll have to finesse the concentration range, but that's straightforward. If your available mass is small, your problem is in getting an accurate value for the mass, and sensitivity of your spectroscopic equipment. The former problem can be solved by using a very accurate balance. At some point the amount of mass becomes so small that an accurate mass isn't so much a problem as does the sensitivity of your spectrometer. You can purchase cuvettes that hold a very small volume of solution, though (as little as 10-20 microliter), which will allow you to amp up the concentration considerably. Keep in mind that for good molar absorptivity determinations you need to acquire spectra at several different concentrations and do a linear regression.
If you don't know the molecular weight, your problem has nothing to do with the molar absorptivity measurement itself. Your problem is finding a molecular weight of your unknown. There are a lot of ways to go about this, depending on the nature of the compound and how much you know about it. If you know the basic molecular weight of your unknown but it is contaminated or contains hydrate or some such, TGA may be an easy way to determine the % by mass that your sample is the actual active substance you are trying to measure. Hydrates are often very easy to process using this technique, particularly if it's an inorganic hydrate.