  A graph is $H$-free if it has no induced subgraph isomorphic to $H$.
Brandst\"adt, Engelfriet, Le and Lozin proved that the class of chordal graphs
with independence number at most 3 has unbounded clique-width. Brandst\"adt, Le
and Mosca erroneously claimed that the gem and the co-gem are the only two
1-vertex $P_4$-extensions $H$ for which the class of $H$-free chordal graphs
has bounded clique-width. In fact we prove that bull-free chordal and
co-chair-free chordal graphs have clique-width at most 3 and 4, respectively.
In particular, we find four new classes of $H$-free chordal graphs of bounded
clique-width. Our main result, obtained by combining new and known results,
provides a classification of all but two stubborn cases, that is, with two
potential exceptions we determine all graphs $H$ for which the class of
$H$-free chordal graphs has bounded clique-width. We illustrate the usefulness
of this classification for classifying other types of graph classes by proving
that the class of $(2P_1+P_3,K_4)$-free graphs has bounded clique-width via a
reduction to $K_4$-free chordal graphs. Finally, we give a complete
classification of the (un)boundedness of clique-width of $H$-free weakly
chordal graphs.
