Dear mrplow,
Thanks for replying. I realize I did not say enough about it. The line appears about 3 ½ cm. above the bottom edge of my LG monitor , and below it appears the top of the current page, flickering on and off. The cable is securely connected, both ends. In the monitor it is indicated to be a D-Sub "head", and in the instructions it apparently is connected by a DVI-D cable. The display connector to my late 2012 Mac Mini (2.5 GHz) is a Moshi and has been working fine.
The monitor is an LG LCD model E2241S which I bought for my 2005 Mac mini, connected since August 2011. I bought the presentMac Mini (2.5 GHz) and the VGA adapter in Feb. 2013.
I don't think the problem is due to electronic things around on my desk (router, small speakers, all-in-one printer )-- nothing has changed. There are transmission towers on a hill in front of us which may interfere, but as far as I know never have done this. We did have an electrical storm I think some time before this happened, but unfortunately I didn't really draw any connections at the time. MacMini and printer are in ethernet LAN with husband. A disc player has rested on the top of the Mini, but hasn't had any problems. However, the DVDs have the same screen problem now; other than that they play well.
I can work on the computer by shortening the height of windows and not using the bottom space of the desktop.
I noted before that when I boot up and close the computer, I always, since the beginning with this Moshi (back in February), had a screen message that says "check signal cable." I called the Mac repair about it, but they said "if it works, don't bother". Up till now there has been no problem, but the message persists.
The flickering band is there all the time.
Carol