Thanks. I'm using the single-transistor interface that they show in the standard application circuits, except using a 2N2222A instead of the one they show -- can't remember why I chose that, but it's worked satisfactorily for me thus far.
Unfortunately, so far as modern electronic devices are concerned, my experience only goes as far as discrete transistors and other basic components plus, of course, the PICAXE itself. So, what is an op amp? From the leaflet in your link, Paix, each of the four sections of it seems to behave exactly like a single transistor, and requires the same number of external components. Is it used simply because of the convenience of having multiple devices in one package?
It seems to be similar to the "Darlington Driver IC" shown in the PICAXE interfacing circuits pages, except the Darling Driver includes the associated base-load resistors etc.
But I digress! Back to my original question ... I'll be trying out the PWM command as soon as I get a chance to have a session in the workshop, and see how it works with my transistor-driven led. I think I understand the way they've used PWMOUT and PWMDUTY in the example in the manual, so I can edit their basic program on p.167 to achieve what I want.
Edit: I've copied the brief program, and edited it to include more steps from 150 down to 0, and have got the basic effect I was wanting. Thanks to all of you for pointing me in the right direction.