I believe it should be possible. I'd start by creating one logfile per user (as I'm unsure of the side effects of sharing a pooled history file).
So for the sake of example, I'm going to
chown root:soneil /var/log/history/soneil
chmod 660 /var/log/history/soneil
So I have a history file that's owned by root, but 'soneil' can write to.
Then, a little magic:
chattr +a /var/log/history/soneil
Now 'soneil' can only append to history, it's otherwise immutable to all but root.
So I've got my log file prepped, I just need to use it.
in /etc/bashrc (on Ubuntu I notice this is /etc/bash.bashrc):
The readonly builtin is fairly self-explanatory, and I believe could be equally applicable to SvenW's function too.
Tested, this has the same problem as the normal history file; it's written at logout, and isn't timestamped. so ordering events would be messy. I think if I had to do this myself, I'd add
to force history to be flushed to disk each time a new prompt is drawn. There's also a
HISTTIMEFORMAT envar which will add timestamps to the history file, but in a rather non-pretty format (setting the var to a prettier format only affects the output of 'history', not the contents of the file itself).