How should I handle age verification in signups?

I have a website where the minimum sign up age is 13 years old. In my sign up form there's a datepicker for selecting their age. The user would enter their date of birth, if they turn out to be below 13 years of age, they receive an error message such as:

You must be at least 13 years old to sign up to this website.

However, after seeing this error message they could easily change their date of birth to something over 13 and sign up anyway.

To circumvent this, I could lock the age field after it's first 'validated', but this would cause other issues. (What if they accidentally entered the wrong date of birth?)

So how should I handle age verification in sign up forms?