Let's assume that
your client's new domain is client.com
your server's ip address (where you have site1.com/newsite) is 192.168.1.1
the path to newsite on your server is /srv/www/newsite
your webserver is Apache
First, you need to create a virtual host in Apache, containing at least:
If you're on Debian based systems, you can just create a new file clientsite.com in /etc/apache/sites-available and put that inside it.
Then run a2ensite client.com and service apache2 reload and you're done.
If you're not on Debian, look on how to add a vhost to your specific distribution apache (CPANEL also has a vhost section for Apache).
Now in your client's registrar, they need to point client.com and www.client.com to your server's ip address 192.168.1.1
Additional: as mentioned in the previous reply, you should prevent access to the /newsite dir so people can only access it by the client.com address, and not by site1.com/newsite.
To do that, in your site1.com vhost, put this
Deny from all
Update the question with more information (include at your OS and webserver and if you have CPANEL or anything similar). The more precise your question is, the more precise the replies you get will be.