How to change the path of the session cookie ASP.NET_SessionId

[svilst]

How can I change the path of the cookie 'ASP.NET_SessionId'?
The path is "/" and I'd like to set the path to particular virtual
directory.

The following approach does not work:

protected void Session_Start(Object sender, EventArgs e)
{
HttpCookie cookie = Response.Cookies["ASP.NET_SessionId"];
if (cookie != null)
{
cookie.Path = Request.ApplicationPath;
}
}

Because an http cookie is idenitified by name,path and domain combined,
changing only the path effectively creates another cookie. As a result
different strange errors occur on the site. So I believe the above is
not quite right.

Does anybody know whether the path of this particular cookie can be
changed programatically or via configuration settings.

Thanks,
Svilen Stoilov