Stop multiple attendance at same event by person

[Simon Finch]

I'm logging attendance at events. I have the following fields;

event_register_id (PrimaryKey)
event_id
person_id

event_id links to another table with the list of events.
person_id links to another table with the list of people.

I'm trying to set up the table so that an event can only be appear once in the table (by using a unique index) as it only happens once on a given date. A person can attend many events but NOT the same event twice.

I'm struggling with how to set this up. I tried a joint unique index with event_id and person_id in the index but that wouldn't allow me to add a person more than once to the table.

Where am I going wrong please?