dWe consider the spin angular momentum evolution of the accreting components of Algol-type binary stars. In wider Algols the accretion is through a disc so that the accreted material can transfer enough angular momentum to the gainer that material at its equator should be spinning at breakup. We demonstrate that even a small amount of mass transfer, much less than required to produce today's mass ratios, transfers enough angular momentum to spin the gainer up to this critical rotation velocity. However the accretors in these systems have spins typically between 10 and 40 per cent of the critical rate. So some mechanism for angular momentum loss from the gainers is required. Unlike solar-type chromospherically active stars, with enhanced magnetic activity which leads to angular momentum and mass loss, the gainers in classical Algols have radiative envelopes. We further find that normal radiative tides are far too weak to account for the necessary angular momentum loss. Thus enhanced mass loss in a stellar wind seems to be required to spin-down the gainers in classical Algol systems. We consider generation of magnetic fields in the radiative atmospheres in a differentially rotating star and the possibility of angular momentum loss driven by strong stellar winds in the intermediate-mass stars, such as the primaries of the Algols. Differential rotation, induced by the accretion itself, may produce such winds which carry away enough angular momentum to reduce their rotational velocities to the today's observed values. We apply this model to two systems with initial periods of 5 d, one with initial masses 5 and 3 M(circle dot) and the other with 3.2 and 2 M(circle dot). Our calculations show that, if the mass outflow rate in the stellar wind is about 10 per cent of the accretion rate and the dipole magnetic field is stronger than about 1 kG, the spin rate of the gainer is reduced to below breakup velocity even in the fast phase of mass transfer. Larger mass loss is needed for smaller magnetic fields. The slow rotation of the gainers in the classical Algol systems is explained by a balance between the spin-up by mass accretion and spin-down by a stellar wind linked to a magnetic field.