I went out and bought a new handset (the old one was rather crappy and we needed one with a better loudspeaker anyway), but the problem remained, so clearly it's something to do with the line.
I called Belgacom's customer service - after the usual insane delving through their website to find the number - and after 20 minutes on hold finally spoke to a technician. He took my details, said he would look into it, and promised to call me back on my mobile number in a few minutes. Needless to say, he never did.
I called them again this morning before leaving home to find out what had happened to my report from yesterday. Needless to say, the technician I spoke to (and I wish I had taken his name; I will always do that with Belgacom staff in future) had made no record of my call. That really is the last straw.
Not being in the office yet, I couldn't give the technical details needed. I got them to promise to call me back mid-morning; but I am not holding my breath.
ETA: In fairness, they did call mid-morning and appear to have fixed the problem! Yay! But I am still unimpressed with their failure to do anything about it yesterday.
Meanwhile I am looking at the other service providers I can go to. If Telenet now cover Brussels (they didn't last time I checked) they will get my business; they have been perfectly acceptable providers at home.