A couple of thoughts:
1) From the picture (may not be true) it looks like the light is much smaller than the marquee - it looks like the middle of the marquee is too bright, and by the right/left sides it isn't lit enough (too dark)? If so, get a florescent light as long as the marquee so the whole thing will be an even brightness
2) As for the middle being too bright - most of the earlier replies are right on, get a light diffuser of some sort (either a real plastic light diffuser, or paper, etc.). My marquee is printed on backlit film, and when I lit it up from the back originally I had the same issue - it was washed out. I had extra backlit film attached to the printed marquee (from Kinkos) so I cut a second sheet out, same size as the marquee (blank part, just white backlit film) and placed that behind the marquee and now it looks perfect. This is the same idea as a diffuser or paper.
Good luck