And there we are. All back up and running.
I attached the new drive to the Mac by a spare SATA external USB case I had, and used SuperDuper to copy the old drive to the new drive. It's automagically made bootable in that process.
Once that had completed, I switched it all off and let it cool down, then set about cracking open the laptop. I followed the iFixit instructions to the letter, fitted the new drive without breaking anything or losing any screws (though one did take a dive for the carpet early on), put it all back together and powered it up.
Amazingly, everything worked as if nothing had changed save the hard drive had acquired an extra 350GB of space.
Thanks for the advice, gents. Much appreciated.

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Snaptophobic BloggageHeather Kay: modelling details that matter. "Let my windows be open to receive new ideas but let me also be strong enough not to be blown away by them." - Mahatma Gandhi.