Ok, forward me your peer-reviewed references, and I'll think about avoiding it.
This is the sensible approach. Before running off in hysterics about the effects of Nutrasweet and other artificial sweeteners, it is best to look at the evidence.

"My mum told me", "I heard", "Nine News told me", "I got an e-mail with unreferenced claims" are all red flags that should lead one to question the validity of what is claimed.

The original article appears to have been circulating since 1995 (refer: http://www.dorway.com/betty/environ.txt)

A quick Google search will bring up a number of sites proclaiming the evils of Aspartame. It should be pointed out that just because something is written on a website, it does not make it true. I'm sure that you could do a Google search and come up with a number of site expounding the virtues of paedophillia as well...

One site lists (with formatting ability of a 4 year old) a number of journal article referring to adverse effects of aspartame. The first article listed is a 2007 article from the European Journal of Clincal Nutrition, whose abstract states:
The aim of this study was to discuss the direct and indirect cellular effects of aspartame on the brain, and we propose that excessive aspartame ingestion might be involved in the pathogenesis of certain mental disorders (DSM-IV-TR 2000) and also in compromised learning and emotional functioning.
I haven't read the full article, so I don't know what the researchers defined as excessive, nor do I know what their research methodolgy was. However excessive amounts of most things aren't good for you. Too much aspirin is certainly bad for one's health.

My quick search yeilded two articles that appear to have some evidence behind them. They are (including the abstracts):

Magnuson BA; Burdock GA; Doull J; Kroes RM; Marsh GM; Pariza MW; Spencer PS; Waddell WJ; Walker R; Williams GM. (2007) 'Aspartame: a safety evaluation based on current use levels, regulations, and toxicological and epidemiological studies', Critical reviews in toxicology, 37(8): 629-727.

Abstract:

Aspartame is a methyl ester of a dipeptide used as a synthetic nonnutritive sweetener in over 90 countries worldwide in over 6000 products. The purpose of this investigation was to review the scientific literature on the absorption and metabolism, the current consumption levels worldwide, the toxicology, and recent epidemiological studies on aspartame. Current use levels of aspartame, even by high users in special subgroups, remains well below the U.S. Food and Drug Administration and European Food Safety Authority established acceptable daily intake levels of 50 and 40 mg/kg bw/day, respectively. Consumption of large doses of aspartame in a single bolus dose will have an effect on some biochemical parameters, including plasma amino acid levels and brain neurotransmitter levels. The rise in plasma levels of phenylalanine and aspartic acid following administration of aspartame at doses less than or equal to 50 mg/kg bw do not exceed those observed postprandially. Acute, subacute and chronic toxicity studies with aspartame, and its decomposition products, conducted in mice, rats, hamsters and dogs have consistently found no adverse effect of aspartame with doses up to at least 4000 mg/kg bw/day. Critical review of all carcinogenicity studies conducted on aspartame found no credible evidence that aspartame is carcinogenic. The data from the extensive investigations into the possibility of neurotoxic effects of aspartame, in general, do not support the hypothesis that aspartame in the human diet will affect nervous system function, learning or behavior. Epidemiological studies on aspartame include several case-control studies and one well-conducted prospective epidemiological study with a large cohort, in which the consumption of aspartame was measured. The studies provide no evidence to support an association between aspartame and cancer in any tissue. The weight of existing evidence is that aspartame is safe at current levels of consumption as a nonnutritive sweetener.
Butchko HH, Stargel WW, Comer CP, Mayhew DA, Benninger C, Blackburn GL, de Sonneville LM, Geha RS, Hertelendy Z, Koestner A, Leon AS, Liepa GU, McMartin KE, Mendenhall CL, Munro IC, Novotny EJ, Renwick AG, Schiffman SS, Schomer DL, Shaywitz BA, Spiers PA, Tephly TR, Thomas JA, Trefz FK. (2002) 'Aspartame: review of safety', Regulatory Toxicolgy & Pharmacology, 35(2 Pt 2):S1-93.

Over 20 years have elapsed since aspartame was approved by regulatory agencies as a sweetener and flavor enhancer. The safety of aspartame and its metabolic constituents was established through extensive toxicology studies in laboratory animals, using much greater doses than people could possibly consume. Its safety was further confirmed through studies in several human subpopulations, including healthy infants, children, adolescents, and adults; obese individuals; diabetics; lactating women; and individuals heterozygous (PKUH) for the genetic disease phenylketonuria (PKU) who have a decreased ability to metabolize the essential amino acid, phenylalanine. Several scientific issues continued to be raised after approval, largely as a concern for theoretical toxicity from its metabolic components--the amino acids, aspartate and phenylalanine, and methanol--even though dietary exposure to these components is much greater than from aspartame. Nonetheless, additional research, including evaluations of possible associations between aspartame and headaches, seizures, behavior, cognition, and mood as well as allergic-type reactions and use by potentially sensitive subpopulations, has continued after approval. These findings are reviewed here. The safety testing of aspartame has gone well beyond that required to evaluate the safety of a food additive. When all the research on aspartame, including evaluations in both the premarketing and postmarketing periods, is examined as a whole, it is clear that aspartame is safe, and there are no unresolved questions regarding its safety under conditions of intended use.
As with the first article mentioned, I haven't accessed the full text of the articles, so can't comment on their methodology. It should be noted that the authors of the third article appear to have been employed by Medical and Scientific Affairs at The NutraSweet Company at the time of publication - this does raise the question of conflict of interest.

I should point out that I personally find that I get headaches when I drink too much soft drink containing artificail sweetener, and that my mate's wife was sick with a differential diagnosis of MS that turned out to be a reaction to too much Aspartame, but my personal experience and that of of my friend's wife does not make it true for everyone.

The bottom line? For me, I try to avoid too much artifical sweetener, but that is my choice. If you are worried, ask your GP when you next have a check up, or look for the evidence yourself.