The Woodwind.org clarinet pages is a great resource for this information.
"A Source of Information for the Clarinet Amateur, Professional, and Anyone
Who 'Just Wants to Know'"
They have a area full of interesting messages called "The Clarinet BBoard."
"A. Fontaine ... mainly sold "stencil" instruments, that is, made by others
but with the Fontaine name on them."
More on what a "stencil" instrument is in this message.
"A. Fontaine was an importer, not a manufacturer. Clarinets labelled 'A.
Fontaine' and "made in France" may have been made by Thibouville Freres or
Couesnon ... Most of the clarinets labelled 'A. Fontaine' that I've seen
were from the WWII to 1950's time period, as best I could tell."
Here, we find out about the "1638" marking. Clarinets were not invented until
1690, it can't be the year it was made, but rather it's a serial number.
"A. Fontaine was a trademark for U.S. imported woodwind instruments. It was
made in one of the many factories in France and stamped with that name."
"On the brass instruments by Couesenon (Couesenon is very famous in the brass
wind trade) the last 2 digits of the serial number indicate the year of
manufacture; no mention is made of how they marked woodwinds."
If the serial number of your clarinet follows this pattern, it might have been
made in 1938.
Your clarinet, highly depending on condition and the actual maker, might be
worth up to $350 or more. From the Marsh Woodwinds web site.
"A. Fontaine Clarinet #2xxx - Wood - in original tweed case - Price: $350.00"
The above is dated to the 1950s. If you click on the link marked "View more
pictures of this item" on the above page, they have an image of the logo.
It might also be worth $150 or less. From the Vintage Instruments web site.
"Clarinet in B-flat, A. FONTAINE BY COUESNON, Paris ... in very good condition,
requires standard overhaul ... $150"
If you need any clarification, feel free to ask.
Google search on: clarinet "A. Fontaine"
Did further searches on the Woodwind.org site for Fontaine.
Looking Forward, denco-ga - Google Answers Researcher