A holomorphic motion of
E
⊂
C
E \subset \mathbb {C}
over the unit disc
D
D
is a map
f
:
D
×
C
→
C
f:D \times \mathbb {C} \to \mathbb {C}
such that
f
(
0
,
w
)
=
w
,
w
∈
E
f(0,w) = w,w \in E
, the function
f
(
z
,
w
)
=
f
z
(
w
)
f(z,w) = {f_z}(w)
is holomorphic in
z
z
, and
f
z
:
E
→
C
{f_z}:E \to \mathbb {C}
is an injection for all
z
∈
D
z \in D
. Answering a question posed by Sullivan and Thurston [13], we show that every such
f
f
can be extended to a holomorphic motion
F
:
D
×
C
→
C
F:D \times \mathbb {C} \to \mathbb {C}
. As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets
C
∖
f
z
(
E
)
,
z
∈
D
)
\mathbb {C}\backslash {f_z}(E),z \in D)
. The proof uses earlier results on the existence of analytic discs in the polynomial hulls of some subsets of
C
2
{\mathbb {C}^2}
.