Magnetic braking is essential for angular momentum transport in late-type stars which have convective envelopes. The mechanism may be entirely responsible for the slow rotation of the present Sun, on which a braking model is normally calibrated. Recent or current satellite missions such as Helios and Ulysses have jointly revealed a more complete picture of the solar corona, and more specifically of the solar wind. The wealth of these data at or near the solar minimum is valuable for constraining the dipolar solar braking model. In this paper, we use recently available observations (at or near the solar minimum) to constrain a solar magnetic braking model based on a dipolar field structure. It is found that the Ulysses data indicate a spherical Alfvén surface at high latitudes. We infer from a thermal wind model that it is located at 16 Ro˙, which is larger than the 12 Rodot; deduced from the Helios data for the equatorial region near the solar minimum. It is also found that the braking model with a transition from a dipole to a split monopole field is generally consistent with Ulysses observations, provided that a linear relation between dead zone extent and dipole field strength is satisfied. Thus either the dipole field retains a sizeable dead zone but is much stronger than the standard value, ∼ 1 G, or the field has standard strength but an exceedingly small dead zone (<2 R x o˙). The magnetic braking rate as constrained by Ulysses data is found to be 2.1× 1030 dyn cm, which is about a quarter of the value deduced earlier from the Weber & Davis model and does not differ significantly from that deduced by Pizzo et al.