As others mentioned, you probably only need to replace the bulb. You have perfectly described what happens when the bulb burns out.
To change the bulb, gently pry the clock out of the dash. You might be able to pry it out with only your finger nails, but you may need a small thin bladed screwdriver. On a Mystique, it is a little harder as you need to remove the dash bezel to gain access to the clock (except for the lattest Mystiques, I think 99 and 00, which are the same as the Contour). The same screwdriver should help remove the bulb and socket from the back of the clock. The socket with the bulb attached twists out like turning the head of a screw.
When you change the bulb, don't forget to transfer the green sleeve covering the bulb. (If equipped, some may not have come with the green sleeve.) If you don't, the clock will be brigher than before and the color will not match the rest of the instrumentation. The sleeve will come off, but it sometimes takes some patience to get it loose.
Last I knew, you could only get the bulb through Ford. It is a unique size.
