How can I update an existing record to have a new auto_increment id in MySQL?

I have a table with primary key (its name is "id") defined as auto_increment. I use NULL in INSERT statements to "fill" the id value. It works, of course. However now I need to "move" an existing record to a new primary key value (the next available, the value is not so much important, but it must be a new one, and the last one if ordered by id). How can I do it in an "elegant" way? Since the "use NULL at INSERT" does not work too much with UPDATE:

update idtest set id=NULL where id=1;

This simply makes the id of the record zero. I would expect to do the same thing as with INSERT, but it seems my idea was incorrect.

Of course I can use "INSERT ... SELECT" statement, then a DELETE on the old one, or I can use something like MAX(id) + 1 to UPDATE the id of the old record in one step, etc, but I am curious if there is a finer solution.

Also, the MAX(id) solution doesn't seem to work either by the way:

mysql> update idtest set id=max(id)+1 where id=3;
ERROR 1111 (HY000): Invalid use of group function
mysql> update idtest set id=(select max(id)+1 from idtest) where id=3;
ERROR 1093 (HY000): You can't specify target table 'idtest' for update in FROM clause

Solution

The way you are trying to update same table is wrong but you can use join on same table

update idtest t
join (select id +1 as id
    from idtest order by id desc 
    limit 1) t1
set t.id=t1.id
where t.id=3;

update idtest t
join (select max(id) +1 as id
    from idtest ) t1
set t.id=t1.id
where t.id=3;