I'm sorry for misunderstanding the intent of your question. Ok, so despite the fact that it violates the intent of constants, you're looking for a way to suppress any error message anytime a constant is defined again with the "use constant..." directive. Here's some rope:

use warnings;
use strict;

BEGIN{ 
    open OLDERR, '>&', \*STDERR 
        or die "Couldn't save STDERR\n$!";
    close STDERR;
}
INIT{
    open STDERR, '>&', \*OLDERR 
        or die "Couldn't retrieve STDERR\n$!";
}

use constant PI => 3.14;
use constant PI => 1000;
print PI, "\n";
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You are now suppressing any error messages that would occur during the stage of compilation where 'use' directives are being sorted out and compiled. That means there are a lot of other errors you won't see. In fact, I doubt you would even see any error caught by my 'or die...' clauses while saving away STDERR. I did give STDERR back to you for runtime stuff though. But this whole thing seems like a BAD IDEA.

By the way, holli is wrong about being able to wrap it in an eval BLOCK. I tried that too. ;)