Maybe if you create a weewx.conf file in /etc/apache2/sites-available and call it weewx.conf
Put this in the file:
Alias /weewx "/home/weewx/public_html"
<Directory "/home/weewx/public_html">
Options Indexes MultiViews
AllowOverride None
Order allow,deny
Allow from all
Require all granted
</Directory>
Save the file.
Then run sudo a2ensite weewx.conf (this enables the site)
Then reload apache