Just had the same problem on my 1999 2.5 CDX.
Over the past few months the starter would randomly not start at the first turn of the key. You could hear a loud click so I guessed the solenoid was playing up but it would fire up on the second or third turn of the key so it never left me stranded. So I decided to ignore it.
The car battery is brand new and fully charged (regularly trickled charged using my CTEK battery charger/conditioner).
However, once the cold weather arrived this started to get worse untill a complete failure last week. No amount of key action would get it to fire. I tried the time honoured trick of hitting the solenoid/starter.....nothing. Luckily the car was stranded outside my house.
So, I replaced the starter and the engine now fires up as normal. Result. I guess 19 years service from the original starter is not bad
Replacing the starter was a bit of a pain (a good article in the maintenance section helps) as I am not as young as I used to be.
I have managed to loose one of the engine mount nuts into the sub-frame!!!! Anyone got any spares? Engine mount lower nut.
The big problem was getting the car off the street and onto my drive to do the work. It's a big car to tow and with no power steering. The wife was not happy!
Now my other Omega (1998, 2.5 Elite) is starting to play up with the exact problem.