Apache/PHP sendmail FROM issue

I've gotten my local Apache server and PHP to work with sendmail. But there is one problem. First, here is my PHP script:

<?php
$to = "myEmail@gmail.com"; // ***** LET'S SAY THIS IS MY EMAIL ADDRESS
$subject = "Testing from local PHP at " . strftime("%T", time());

$message = "Hello, PHP world!";

$from = "no-reply@localhost"; // ***** THIS IS THE 'FROM' EMAIL ADDRESS I SPECIFIED
$headers = "From: {$from}";

/**
 * The mail() function returns true or false depending on whether it was successful
 * in sending it.
 */
$result = mail($to, $subject, $message, $headers);
echo $result ? "Sent" : "Error";
?>

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The email succeeds, but the FROM always has myEmail@gmail.com, not the no-reply@localhost I specified. Can anyone tell me how to correct this? There aren't really much to configure in sendmail.ini, and I even did:

force_sender=no-reply@localhost.com

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I can't see anything else I can modify that might affect the FROM field. Anyone have any advice? Thanks.