Motobécane was one of the premium builders of French bicycles in the 1970s, and along with Peugeot they seem to have been well known for their mixte frames. They were known for using Vitus and Reynolds 531 for their mid to high end frames. Mine says 'built with 1020 tubes and stays' which I've always taken to mean it's made of a less expensive cro-moly. It's light but not super light, and certainly tubing and not pipe steel. I've never actually come across any conclusive information about 1020 tubing, so this is just assumptions based on the weight and quality of the frame, and the lack of any manufacturer branding of the tubing.
They are also noted for beautiful and high-quality paint, which my example certainly confirms. Over 30 years later, the paint is not so beautiful, but it is all still there, in itself a testament to quality.
These bikes make excellent fixed gear conversions, with a few caveats. The biggest complications to be faced in such a conversion relate to the weird threading of the headsets and bottom brackets, which were unique to the French bike industry.
With the headsets, other 1 inch headsets may fit, but to use the original fork with the frame, you'll need an original French headset. I got lucky; the headset on this bike took some rubbing out with steel wool and fresh bearings and grease, and was like new. A lot of these parts can be reconditioned and re-used with a little polishing of the bearing races. Look out for deep pits or scars that may interfere with the smooth rolling of the bearings.
The bottom brackets are likewise threaded unusually. They are not interchangeable with bikes from other countries, or most new bike parts, which use the now-standard English threadings. Because you will be limited to using 30 year old cups and crank axle, you'll be limited to the square tapered bottom bracket style of crank arm fitting. This was in common use on performance road bikes of the time, and single speed specific cranks are widely available in many grades. I'm using an Origin8 crank from J&B, you can order it at your LBS. I got lucky in Lafayette, IN. The old road crank I'd been using collapsed on my first day in town, and it turned out I was right around the corner from Hodson's Bay, the best LBS if you're in that county. They had the part I needed on the shelf.
When I orginally built this into a fixed, it took a bit of digging around in parts bins to find a crank axle that was right. It needed to be the correct width for a decent chain line; it also needed use the square taper fitting rather than the older, cheaper cottered design the bike originally featured. Again, some steel wool was used to rub out any corrosion in the cups and races. There is no hope of fitting a more modern sealed bearing design, if one was so inclined. However, the old one's should be possible to keep in service indefinitely with a little preventative maintenance.
Of course, these are also frames for 27" wheels, not the more common 700c, so finding a brake to reach the front wheel correctly may take some experimenting. One will need a source with a bunch of old 70s era performance bike brakes to find one with the right reach.
Finding a modern seat post was fun too. This involved measuring to the nearest larger size and then reaming out the seat tube a bit on my bike.
If these problems don't scare you off, and even seem like part of the fun, then converting an old French bike to fixed can be a relatively inexpensive path to a unique bike with history and style all it's own. Otherwise it may be best to stick with modern parts or to start with a new factory built fixed gear. But where's the fun in that?