How to split a string into an array in Bash

Question

In a Bash script, I would like to split a line into pieces and store them in an array.

For example, given the line:

Paris, France, Europe

I would like to have the resulting array to look like so:

array[0] = Paris
array[1] = France
array[2] = Europe

A simple implementation is preferable; speed does not matter. How can I do it?