A dispensing container for wet wipes includes a pop-up style dispensing means formed by a rigid port which surrounds a flexible, rubber-like material or sheet having one or more slits through which the wet wipes are dispensed. The configuration of the slits, in combination with the flexibility of the rubber-like material, enables the user to insert at least one finger through the slit to thread a wipe through the slit. The dispensing means has a Penetration Index of greater than 600 grams.