Dhampir bite attack, does it add both STR and CON? (DND 5e)

Here’s a little extract from the Dhampir bite attack (emphasis mine):

"Your fanged bite is a natural weapon, which counts as a simple melee weapon with which you are proficient. You add your Constitution modifier to the attack and damage rolls when you attack with your bite. Your bite deals 1d4 piercing damage on a hit. While you are missing half or more of your hit points, you have advantage on attack rolls you make with this bite."

Do I add both STR and CON to it? Or does CON replace STR?

I’m a bit confused, because it counting as a simple melee weapon attack means it would normally be 1d20+STR+prof to hit and dmg_die+STR to damage. The wording is pretty different from, for example, stuff like Finesse weapons, which makes me think it adds both modifiers. Which makes the attack roll something like 1d20+STR+CON+prof and the dmg roll 1d4+STR+CON. Is that correct?