We determine the critical coupling of the two-dimensional Brown–Ravenhall operator with Coulomb potential. Boundedness from below has essentially been proven by Bouzouina. However, that work contains a trivial error leading to an incorrect constant that is exactly half of the actual critical constant. Furthermore, we show that the operator is in fact positive. Our proof of that is, for the most part, analogous to Tix's proof of the corresponding result for the three-dimensional operator.