So far as any scientific study has shown, there's nothing in Aspartame that poses any danger to humans... except its taste.
Any time you ingest something that tastes strongly of sugar, your body (prompted by the brain) releases insulin. If there's no sugar in your stomach at the time, and particularly if there's no sugar there and your liver's supply of glycogen (sugar stored for quick release) is depleted (say at the end of a long day), you're risking insulin shock - the mechanism behind diabetic seizures, which occur when diabetics inject insulin and then forget to eat some carbohydrates. So it shouldn't be big surprise that despite repeated studies showing Aspartame itself to be safe (and actually safer as your body gets more used to it), perhaps the most common claim against it on the internet is that “it causes seizures”.
Particularly if your body isn't used to aspartame, and hasn't learned that you like to tease it this way; add a little sugar (preferably longer acting fructose) to the drink. Otherwise, you'll be a good candidate for a mild seizure such as an absence seizure, and if you're subject to migraines, one of those isn't exactly unlikely either. (You could just eat, say, a couple of corn chips with the pop, of course, but you might not stop at just two.)
Fructose (fruit sugar) is absorbed slowly and won't act as an appetite stimulant, so you'll still have something of a diet drink (assuming you add only a little). In contrast to fructose, studies show sucrose stimulates the appetite by raising and then crashing blood sugar levels. We haven't evolved to handle sucrose well because it is extremely rare in nature: only a handful of plants produce it in any quantity.
Needless to say, the above cautions apply to any and all sugar substitutes which might cause the release of insulin in the absence of actual carbohydrates. (Some diet substitutes for sugar, such as mannitol, are actually sugars, but less caloric.)
It seems likely, or at least possible, that over a long time of exposure to Aspartame your body, so to speak, would learn to be more cautious in releasing insulin, presumably making seizures far less likely. However, if it therefore began to release less insulin in all sugary situations, this adaptation could produce powerful side effects itself - perhaps greatly increasing blood sugar levels since sugars weren't being stored, or perhaps a strong dietetic effect, as fewer sugars were being stored from meals or snacks. Needless to say, more study is needed here.
If you're diabetic, and don't have a significant insulin response, aspartame should be perfectly safe for you.