If this file can get large, you may get some benefit out of not rewriting the whole file.

#!/usr/bin/perl
my $file=shift(@ARGV);
open(FILE,"+<$file") or die;
while (<FILE>) {
    if (eof(FILE)) {
       seek(FILE,-(length($_)),2) or die("$!\n");
       print FILE "replacement string\n";
       last;
    }
}
close FILE;
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Note that you don't really have to read the whole file either. You could seek() to the end and then back up looking for a newline. Or, if each line is the same length ( it appears that they may be...), you can just go the end and back up X bytes.