CURSOR Is Driving Me Insane!!

We have a tree structure containing section names.  Each node is a
section name and each section can have subsections.  I have to copy the
tree structure but need to maintain the parent-child relationship
established within the id / parent_id fields.  How do i acheive this?

For example i have the tree
Section 1
   |-Section 1.1
Section 2
   |-Section 2.1

The
"Section" table contains 3 fields: id, parent_id, and caption.  ID is
the identity of the section record and parent_id contains NULL or the
ID of this record's parent to create a child.  So "Section 1" (id=1,
parent_id=null), "Section 2" (id=2, parent_id=null), "Section 1.1"
(id=3, parent_id=1), "Section 2.1" (id=4,parent_id=2).

I would
like to copy this sucture to create 4 new sections but they need to
maintain their id/parent_id relationships BUT with new IDs.  For this i
created the following stored procedure:
----------------
CREATE PROCEDURE [dbo].[CopySection]
AS
     -- Declare a temporary variable table for storing the sections
     DECLARE @tblSection TABLE
     (
          id int,
          parent_id int,
          caption varchar(max),
     )

     DECLARE @newAgendaID int, @newSectionID int;
     DECLARE     @tid int, @tparent_id int, @tcaption varchar(max);

BEGIN
     -- SET NOCOUNT ON added to prevent extra result sets from
     -- interfering with SELECT statements.
     SET NOCOUNT ON;

     -- Copy the desired sections into the local temp variable table
     INSERT INTO @tblSection SELECT id, parent_id, caption FROM tblSection ORDER BY parent_id;

     -- Using a cursor, step through all temp sections and add them to the tblSection but note its new ID
     DECLARE c1 CURSOR FOR SELECT * FROM @tblSection ORDER BY parent_id FOR UPDATE OF parent_id;
     OPEN c1;

     FETCH NEXT FROM c1 INTO @tid, @tparent_id, @tcaption;
     WHILE @@FETCH_STATUS = 0
     BEGIN
         
          -- Insert the new Section and record the identity
          INSERT INTO tblSection (agenda_id, parent_id, caption) VALUES (@tparent_id, @tcaption);
          SET @newSectionID = SCOPE_IDENTITY();

          -- Update the temp variable table with the new identity from the newly created real section in tblSection
                -- Update all temp variable records to point to the new parent_id
          UPDATE @tblSection SET parent_id = @newSectionID WHERE parent_id = @tid;

          FETCH NEXT FROM c1 INTO @tid, @tparent_id, @tcaption;
     END

     CLOSE c1
     DEALLOCATE c1
     
END
----------------

The
critical "UPDATE @tblSection" part doesnt seem to update the temp
variable table with the @newSectionID (the actual section identity
obtained after inserting a real record into the tblSection table).  So
in the end the inserted records into tblSection still point to the
incorrect parent_id instead of the copied record's parent_id.

Maybe I'm using CURSOR incorrectly or not setting a parameter so that it refreshes its recordset?  I've tried using both table variables and temp tables but no luck.