You've given several solutions, both Perl and non-Perl, but I thought I'd provide yet another short-cut. It uses the nice File::Slurp module. If you care about memory, don't use this method, but use the way described in Re: How can I concatenate two files? instead.

  use File::Slurp qw/ read_file append_file /;

  append_file('A.txt', read_file('B.txt'));
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ihb

Read argumentation in its context!