[Hampshire] tailing a file as/after it gets rotated?

Hi,

I have a log file on a server, which I like to keep an eye on by
running tail -f logfile (in fact I then pipe that through other stuff
too). This sits in a dedicated screen session.

Trouble is the file I am tailing gets moved by a cron job at 06:00
every day and replaced by a new empty one. My tail job keeps watching
the old log file even though it has now been renamed by the rotate
procedure, and the new logfile is untailed.

I am sure this is not a new thing but a (not so) quick google did not
come up with a way round this. All I can think of is to have the cron
job that rotates the log file send a HUP signal to any process tailing
it, but I'm not sure how one would go about that or if that would
actually work. (Having just experimented, I suspect not.)
( I have just noticed that this paragraph includes five occurrences of
the word 'not', and that's not including these two. ;-)

SO, does anybody know of a way to keep a tail on-target when the files
get whiked away and replaced?