A few weeks ago Keni Barwick pointed me at Alarm Master by Connective tools. It runs on the V1240 phone amongst others.
I purchased a copy for less than £5 ($8.95 in US Dollars to be exact) and I'm happy to say that it works for me on my V1240. The program's interface is simple and slick, and it does all that I need from an alarm: I can set it up to wake me Monday to Friday at a set time, and I can set up additional once-off alarms. What more could you want from an alarm? Not much, but I wonder why an alarm with this functionality doesn't come free with the phone in the first place.