Chris,
I hope I'm safe in assuming that this is a CRT monitor... otherwise,
disregard the following:
(1) a vertical line means no horizontal deflection, but
(2) a visible line means there is high voltage on the CRT, meaning the
horizontal oscillator is buzzing merrily along.
Chances are that the horizontal deflection coil on the yoke is open.
If you're lucky, a wire may have broken running to the
electromagnet-thingy clamped on the neck of the tube - if not, getting a
replacement yoke depends on how much support you can get from Apple.
Alternatively, find a similar monitor that died of a different problem
and mug it for the replacement. Chances are that the display will look
distorted, since each yoke has little permanent magnets glued to it that
linearize that particular yoke for its CRT. This is the issue even for
a brand-new yoke.
Bob Maxwell