Not sure about your 2Gb file size problem, but your general solution to the problem seems a little inefficent. You're opening (or reopening) a file for each line of the log file. It would be more efficient to do something like this (in perl/pseudocode mix!):

set $prev_date to an impossible value
while (<>) {
  Get $date from line
  if ($prev_date ne $date) {
    close FILE;
    open FILE, "new_filename_calculated_from_$date"
      or die "$!\n";
  }
  print FILE, $_;
  $prev_date = $date;
}

close FILE;
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European Perl Conference - Sept 22/24 2000, ICA, London
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