Try passing $user_ID global variable to function as following. it is working for me.
<?php
global $user_ID;
if ( get_the_author_meta('user_url',$user_ID) ) : // If a user has filled out their decscription show a bio on their entries
?>
<a class="meta-website" href="https://wordpress.stackexchange.com/questions/89947/<?php the_author_meta("user_url',$user_ID); ?>">Author's Website</a>
<?php endif; ?>