We prove that
E
x
t
A
∙
(
k
,
k
)
\mathrm {Ext} ^{\bullet }_A(k,k)
is a Gerstenhaber algebra, where
A
A
is a Hopf algebra. In case
A
=
D
(
H
)
A=D(H)
is the Drinfeld double of a finite-dimensional Hopf algebra
H
H
, our results imply the existence of a Gerstenhaber bracket on
H
G
S
∙
(
H
,
H
)
H^{\bullet }_{GS}(H,H)
. This fact was conjectured by R. Taillefer. The method consists of identifying
H
G
S
∙
(
H
,
H
)
≅
E
x
t
A
∙
(
k
,
k
)
H^{\bullet }_{GS}(H,H)\cong {\mathrm {Ext}}^{\bullet }_A(k,k)
as a Gerstenhaber subalgebra of
H
∙
(
A
,
A
)
H^{\bullet }(A,A)
(the Hochschild cohomology of
A
A
).