That's false. The article does a good job of obscuring this, but there are far fewer voting than are registered in those counties, and the number voting is less than the population.
I'll list the county, number active registered, number voting, population per census, people actually voting/population in %. The last one is key:
Lowndes, 9585, 7519, 8540 (88%)
Greene, 7,224, 5,328, 6859 (78%)
Macon, 17,071, 10,385, 16,934 (61.3%)
Wilcox, 8,573, 6,547, 8,567 (76.4%)
So, in each county, fewer voted than are listed as the adult population.
Besides, voter fraud leaves a trail. There is a list of everyone who voted. If Alabama thinks there are a large number of the blahs voting twice, they should assign someone to investigate, and find at least ONE case of actual voter fraud. These stories pop up every cycle, and despite the huge advantages to the "photo ID" proponents, we never move from "hey, this is odd..." to cases of demonstrated voter fraud. And the reason is that in person voter fraud is trivial, but bad paperwork and lists that aren't properly updated as common as dirt.