Given a closed surface
X
X
, the covering solenoid
X
∞
\mathbf {X}_\infty
is by definition the inverse limit of all finite covering surfaces over
X
X
. If the genus of
X
X
is greater than one, then there is only one homeomorphism type of covering solenoid; it is called the universal hyperbolic solenoid. In this paper we describe the topology of
Γ
(
X
∞
)
\Gamma (\mathbf {X}_\infty )
, the mapping class group of the universal hyperbolic solenoid. Central to this description is the notion of a virtual automorphism group. The main result is that there is a natural isomorphism of the baseleaf preserving mapping class group of
X
∞
\mathbf {X}_\infty
onto the virtual automorphism group of
π
1
(
X
,
∗
)
\pi _1(X,*)
. This may be regarded as a genus independent generalization of the theorem of Dehn, Nielsen, Baer, and Epstein that the pointed mapping class group
Γ
(
X
,
∗
)
\Gamma (X,*)
is isomorphic to the automorphism group of
π
1
(
X
,
∗
)
\pi _1(X,*)
.