Localise the change to STDOUT. EG:-

{
    local *STDOUT;
    open(STDOUT, "> $LogFile") || die "Failed: $!";
    print STDOUT "Blah";
}
print "This should appear on the console window again";
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However, I do have to ask - why use STDOUT for this as opposed to another filehandle?

Update: Aaarrrggghhh. Beaten to it by BUU. Apologies for the duplicate response.