I find this sort of thing quite amusing really (not refering to the op particularly by the way) " oh my god i've been in contact with asbestos, its dangerous and might make me ill" and yet we manufacture sell and burn millions of "cancer sticks" !!!!
The health risks from something as described in the op's post must be very minimal.
As joe says the asbestos removal companies like to make a big issue of it, don't get me wrong it needs to be treated with respect, but life is full of risks (it would be very boring if we didn't have to take any risks, which seems to be the health and safety police's ultimate target)
Lets get one thing straight - i am not 'hamming it up' and stand to make no profit from the advice. I own one of only 3 types of business which requires a government enforced licence, and as such have a duty of care to the public to inform and warn if necessary. This was such a case.
Everyone swallows asbestos fibres EVERY DAY, its in the air and on the roads and in the schools. I would hazard a guess that 50% of your children are sitting not far from some as i type, but as long as it is well managed there is no risk.
Anyone who has stood on that London Underground platform feeling that rush of air that lets you know the train is coming - you have sucked in copious amounts of BLUE (chrysidolite) asbestos - the worst kind.... so no, i'm not going to 'ham it up' over a broken piece of Amosite.
And no it is not Supalux/Masterboard or any other asbestos alternative. As they say, too little knowlege can be a dangerous thing, and some replies on this thread just enforced that.
Also, i think you will find if the council tip does have an asbestos facility,they will inly accept cement bonded asbestos, i.e white asbestos. Amosite is in a different league and they will not take it (so long as they can see what it is).
Amanda i hope i have been of help.
Regards, Dean