Can the web server be at one hosting company, and the email hosted by a 3rd party

First download the JavaMail API and make sure the relevant jar files are in your classpath.

Here's a full working example using GMail.

import java.util.*;
import javax.mail.*;
import javax.mail.internet.*;

public class Main {

    private static String USER_NAME = "*****";  // GMail user name (just the part before "@gmail.com")
    private static String PASSWORD = "********"; // GMail password
    private static String RECIPIENT = "lizard.bill@myschool.edu";

    public static void main(String[] args) {
        String from = USER_NAME;
        String pass = PASSWORD;
        String[] to = { RECIPIENT }; // list of recipient email addresses
        String subject = "Java send mail example";
        String body = "Welcome to JavaMail!";

        sendFromGMail(from, pass, to, subject, body);
    }

    private static void sendFromGMail(String from, String pass, String[] to, String subject, String body) {
        Properties props = System.getProperties();
        String host = "smtp.gmail.com";
        props.put("mail.smtp.starttls.enable", "true");
        props.put("mail.smtp.host", host);
        props.put("mail.smtp.user", from);
        props.put("mail.smtp.password", pass);
        props.put("mail.smtp.port", "587");
        props.put("mail.smtp.auth", "true");

        Session session = Session.getDefaultInstance(props);
        MimeMessage message = new MimeMessage(session);

        try {
            message.setFrom(new InternetAddress(from));
            InternetAddress[] toAddress = new InternetAddress[to.length];

            // To get the array of addresses
            for( int i = 0; i < to.length; i++ ) {
                toAddress[i] = new InternetAddress(to[i]);
            }

            for( int i = 0; i < toAddress.length; i++) {
                message.addRecipient(Message.RecipientType.TO, toAddress[i]);
            }

            message.setSubject(subject);
            message.setText(body);
            Transport transport = session.getTransport("smtp");
            transport.connect(host, from, pass);
            transport.sendMessage(message, message.getAllRecipients());
            transport.close();
        }
        catch (AddressException ae) {
            ae.printStackTrace();
        }
        catch (MessagingException me) {
            me.printStackTrace();
        }
    }
}

Naturally, you'll want to do more in the catch blocks than print the stack trace as I did in the example code above. (Remove the catch blocks to see which method calls from the JavaMail API throw exceptions so you can better see how to properly handle them.)

Thanks to @jodonnel and everyone else who answered. I'm giving him a bounty because his answer led me about 95% of the way to a complete answer.

If you can configure a reverse proxy... then yes.

Apache mod_proxy

IIS reverse proxy <= is is a codeproject solution and may not have support. You can probably find many more if you just search for reverse proxy.

If you are using IIS7 (Vista, Windows 7, Server 2008) you may use the HTTP Redirect feature that is built in.