Session cookie not expiring...?

[me]

If I start a session on our web app then close all browsers, I expect that
the session cookie would be deleted and that the session itself would exist
until it expired. I then start a new instance of the browser expecting to
get a new session id and a new session, but instead I reconnect to the
unexpired session. Does anyone know why the session cookie is persisting
despite the closure of all the IE windows?

Thanks,
scott.