Judy - the reason that the amber broke is two fold -
It wasn’t annealed, and second, the “amber” from Mexico isn’t really
amber, it is technically copal, or young amber. Gemmologically
speaking, amber is 40 million years old, anything else still has too
much volitile chemistry to be amber.
The most stress in a piece of amber is where inclusions are found, so
amber with insects is usually not annealed. Annealing is done at a
relatively low temperature, about 170 F, in oil. this results in the
characteristic discs, from spreading out the stress. If it is done
with buggy pieces, the discs would cover the bugs, thus removing the
interest and value.
Now to your question, yes you can glue the piece together, being as
the piece is from Mexico, you might try acetone to see if it softens
it. If so, it will make a nearly invisible bond. Acetone is often
used to “polish” copal grade material since it softens the surface.
In the event this doesn’t work, use nail glue - the stuff that comes
in a little tube for fake nails - it has a tiny applicator and
ususally works great. No I don’t do my nails - I work with this
jewelry stuff and haven’t had a manicure for years.
Good luck, I love amber, it’s color, weight, and warmth!
Judy - in steamy hot Denver, with a weeded garden and a sunburn.