Does a table with a surrogate key require a unique constraint on a natural key to be in 1NF?

Question

The pragmatists have won the argument of surrogate vs. natural primary keys in favor of surrogate keys*. In my own work I always use SQL Server identity columns without a second thought. But it occurs to me that, for a table to be in 1st normal form, I should be able to identify a natural key and enforce it with a unique constraint. I can't do this for all the tables in my database, so my database doesn't even meet the lowest criteria of normalization.

Do you agree that a table with a surrogate primary key must also have a unique constraint on a natural key in order to be in 1NF?

*I think Joe Celko is still fighting the good fight, see the last paragraph.

Edited to add: Thanks for the responses. My impression is that adding a unique constraint is not a common practice, so I'm somewhat surprised that the responses so far have been unanimous.

Answer 1

Yes!

If the table is supposed to record at most instance of the natural key, you need a constraint on the relevant column(s) to enforce that. Otherwise, you can end up with a table of 50,000,000 rows, each with a different ID value but otherwise identical -- which is pathetic or ludicrous or a travesty, depending on your viewpoint.