  We provide a spin-adapted formulation of the Full Configuration Interaction
Quantum Monte Carlo (FCIQMC) algorithm, based on the Graphical Unitary Group
Approach (GUGA), which enables the exploitation of SU(2) symmetry within this
stochastic framework. Random excitation generation and matrix element
calculation on the Shavitt graph of GUGA can be efficiently implemented via a
biasing procedure on the branching diagram. The use of a spin-pure basis
explicitly resolves the different spin-sectors and ensures that the
stochastically sampled wavefunction is an eigenfunction of the total spin
operator $\hat{\mathbf S}^2$. The method allows for the calculation of states
with low or intermediate spin in systems dominated by Hund's first rule, which
are otherwise generally inaccessible. Furthermore, in systems with small spin
gaps, the new methodology enables much more rapid convergence with respect to
walker number and simulation time. Some illustrative applications of the
GUGA-FCIQMC method are provided: computation of the $^2F-^4F$ spin gap of the
cobalt atom in large basis sets, achieving chemical accuracy to experiment, and
the $^1\Sigma_g^+$, $^3\Sigma_g^+$, $^5\Sigma_g^+$, $^7\Sigma_g^+$ spin-gaps of
the stretched N$_2$ molecule, an archetypal strongly correlated system.
