It is not known. There are many reports of headaches caused by 'aspartame', but headaches are one of the main complications that arise from the placebo effect as well. In trying to establish whether a difference exists between placebo and aspartame in regards to headaches, only one blinded study has been conducted and it was riddled with errors.
There is not enough evidence to make a final conclusion either way. Use your own judgement, as headaches are something you are aware happens and can avoid if caused by the diet.
Aspartame and Headaches
One of the most common complaints about aspartame reported to the Center for Disease Control (CDC) are headaches. Specifically, "346 (67%) complainants reported neurological/behavioral symptoms, including headaches, dizziness, and mood alterations".
Although a concern to the molecule itself (and some old studies hypothesizing neural effects of aspartame), headaches are also frequently caused by the placebo effect. Due to the notoriety of aspartame, it is theoretical that many of the headaches could be induced from anxiety. The CDC report notes that:
Overall, the 517 interviewed complainants were predominantly white (96%), female (76%), and between the ages of 21 and 60 years (79%). Complainants were approximately 1.5 times more likely to be female and 2.5 times more likely to be women between the ages of 20 and 59 than expected from 1980 census estimates. Reports came from all geographic regions of the country, with a heavy concentration of cases from Arizona, where the possibility of aspartame use leading to illness received particularly extensive press coverage
Which suggests that demographics and the media may come into play due to abnormal demographic dispersion, unless there are genetic differences in Caucasian females from Arizona that are at greatly different rates than the rest of the US.
Studies into Headaches
Despite aspartame's large therapeutic window and general safety, at least one blinded intervention, which only tested those who self-reported aspartame headaches (thus controlling for non-respondants), suggests that aspartame may be more significant than placebo in triggering headaches in some. This particular study has received three official critiques after its publication, due to its sampling and methodology.
Said study was conducted in 1995 and no attempts have been made at replication of the study.
Beyond the above highly questioned study, a few case reports arise sporadically linking aspartame weakly to neural complications, such as increasing a migraine and two twins who appeared to have a genetic response linking aspartame to vestibulocochlear toxicity (hearing).