The spin-boson model is solved within the framework of quantum-classical dynamics using our recently-developed surface-hopping scheme. The quantum-classical equation of motion is expressed in an adiabatic basis and its solution is constructed from an ensemble of trajectories which undergo nonadiabatic transitions and evolve coherently on the adiabatic surfaces. Details of the algorithm for the simulation of the dynamics are presented and the method of simple Monte Carlo sampling used to evaluate the expectation values of observables is discussed. The simulation method is applied to a spin-boson system with a harmonic bath composed of ten oscillators with an Ohmic spectral density. For the spin-boson model the present implementation of quantum-classical dynamics is exact and the results of our surface-hopping simulations are in accord with previous numerically exact results for this model.