There are a few interesting edge cases you may care to take care of. Consider:

use strict;
use warnings;

<DATA>;
print ''.(chomp, s/\s*$/ /, $_).($_ = <DATA>, s/^\s*(?=\S)//, $_) whil
+e <DATA>;

__DATA__
Header line
First line part 1.
Part 2 of first line.
Second line parts one  
  and two.
Third line - second part empty.

Fourth
and last line
[download]

Prints:

First line part 1. Part 2 of first line.
Second line parts one and two.
Third line - second part empty. 
Fourth and last line
[download]

which handles a number of cases, but spits warnings if the last line of the final pair is missing.