You're better off with one of the Motobecanes then the fixed gear. They will give you a wider range of riding opportunities.
One thing to remember is not to go super cheap. It's something my wife figured out this year. We bought her a very cheap bike to see if she would like it. When you would push on the pedal it would move but the bike itself wouldn't. Needless to say it was not easy for her to ride or keep up while I was on skates. This year we got her a much nicer bicycle and she has increased her distance by 1/3 and average speed by about 2mph with less effort.
You can try my bike if you want to come out here. We have a nice bike path. The catch is you will need to have size 9 feet because I have clipless pedals. Probably something you will want to get eventually.
If you don't need new it might be to your advantage to message PK. I know he has a motobecane that he bought for his son that isn't getting much use as far as I know. If its the size you're looking for maybe you can pick that one up.