I found the answer explained here: http://technet.microsoft.com/en-us/library/ms131053.aspx
The connection that the CLR procedure runs from is called the "Context Connection" and it is used like this:
using(SqlConnection connection = new SqlConnection("context connection=true"))
{
connection.Open();
// Use the connection
}
I wonder why you have to "open" the connection? I would think that the connection is already open since it is executing the procedure that is running.
Also closely related: http://msdn.microsoft.com/en-us/library/938d9dz2(v=vs.90).aspx