The most reliable way to determine alcohol in any fermented drink, without the need for several thousand US dollars in equipment and some seriously technical laboratory procedures, is distillation.
The answers already given provide good solutions in certain circumstances, but both will give incorrect readings in solutions with significant amounts of dissolved substances (especially sugar), because these compounds alter the physical characteristics of the liquid which are being used to determine alcohol content. In general, these methods only apply directly to the behavior of pure mixtures of water and ethanol, and when you measure something else (e.g. any fermented drink) other dissolved substances (sugar, protein, organic acids) will skew the results (or, in other words, unless you're measuring a pure mixture of alcohol and water, any reading you get is merely an extrapolation, telling how the liquid you're measuring compares to a pure mixture's behavior).
Distillation, however, ensures that you are only measuring volatile substances and leaving behind non-volatile dissolved constituents. Ethanol and water are the most volatile substances you're likely to encounter in the largest quantities. Acetic acid (vinegar), which may be found in some fermented beverages is also mildly volatile (about half as much as water at distillation temperatures), but its density is very similar to water and can likely be ignored in all but the highest concentrations. Volatile acids (including acetic) may be neutralized with addition of a base (sodium hydroxide), which converts them to a non-volatile form and thus removes the error they may otherwise present.
Other substances are highly volatile but are almost always found in such minute quantities as to make them completely negligible (parts per billion for some).
It's not the most practical measurement to be making for the home fermenter (though you could probably get all the tools you'd need for a lab-grade distillation apparatus for a couple hundred US dollars), but it is the most accurate. Most other methods are still calibrated to distillation.
Here is one outline of a common method, and another. You could also buy one of these if you have a large surplus of money on hand.
For reference, the American Society of Brewing Chemists lists distillation as its primary method for measuring alcohol content.
There are, of course, many other methods used for high-accuracy alcohol determination. Generally, these methods involve pricy equipment and intricate sample preparation and measurement procedure, and for this reason, to me, lie outside of general home fermentation practice:
- Refractometry - the refractive index of a liquid is used along with specific gravity to determine alcohol content. A few issues include:
- Refractometer needs to be read in 'refractive index', not Brix or other commonly found units. Something like this, or any other abbe refractometer.
- This method generally uses a calibration curve that is based on the distillation method (the ASBC method mentions it may be necessary to build a calibration curve for every single type of beverage to be tested).
- There are refractometers for measuring alcohol content directly, but which require the sample to be distilled first.
- Honestly this technique may be well within many home fermenters' limits of budget and patience, but may be inaccurate without a proper calibration curve (see above). Here is a sample procedure.
- Gas Chromatography - you don't have to do much Googling to understand why this is out of the reach of most people. However if you have access to the equipment and methods, it's definitely a good method, even for discriminating and quantifying different kinds of alcohols (i.e. fusel alcohols)
- HPLC (high performance liquid chromatography) - same as above.
- Enzymatic analysis - in which alcohol is reduced by specific enzymes in a controlled environment and the results measured with a spectrophotometer.
- Proprietary methods - such as the Anton Parr Alcolyzer or similar devices.
Ethanol concentration measurement is actually an active field of research, mostly in the pursuit of methods that can be automated and used in-line in large volume plants.
So, while there are many highly accurate methods out there, I still believe distillation to be the best compromise between quality results and being within reach of the average 'highly-motivated' home fermenter.