A trick my mother uses is to weigh it, but this only works in some very select cases and tends not to be conclusive in all but the most extreme cases.
Because some modern soft drinks are up to 1g sugar per 10mL volume, they tend to be a fair bit denser than water. Since aspartame is much, much sweeter than sugar, you usually only need a tiny bit (according to Wikipedia, only 180mg per 12 fl oz) and it doesn't increase the density as much.
Of course, this only has a hope of working if you're comparing very sweet beverages (e.g. Mountain Dew and its 0.13g/mL of sugar), and isn't terribly conclusive on iced teas or lightly sweetened beverages. Still, it's an interesting technique that doesn't rely on trying to chemically separate the two substances.
Note: If you actually do try this, make sure that both beverages are completely flat (decarbonated) and at the same temperature.