Apologies in advance that I have no idea what a 6 year old Dawes Giro 300 has in terms of components, so I shall assume your "old fashioned" adjective meant the headset is threaded (to confirm see
this).
As you can see from Sheldon's article, replacing a threaded headset is not the easiest of DIY bike maintenance projects. Removal of the old and installation of the new can call for special tools (although some folks do use old pipes, hammers etc. to improvise...). Also it is critical that the new one is of the correct
stack height for it to work, so you have to be careful of what you buy.
A Campag Record 1" threaded headset, which is super light and beautifully made,
retails for a surprisingly modest £40. However much of it is made of light alloy, so I would definitely not recommend installing it without the special tools.