An estimation of the solar wind speed in the vicinity of the Sun is carried out using the initial speed and acceleration of coronal mass ejections (CMEs) that appeared close to the solar limb. A linear relationship was found between the initial acceleration and the speed of the limb CMEs. It appears that a dragging force is acting on the CMEs, depending on the speed difference between the CMEs and the ambient plasma. The ambient solar wind speed within 20 solar radii estimated from low-latitude CMEs during 1998--2003 ranged from 100 to 700 km s-1, while the solar wind speed measured at 1 AU ranged from 300 to 700 km s-1. The estimated solar wind speeds in the vicinity of the Sun sometimes agreed with the simultaneous in situ measurements at 1 AU, but in other periods they were slower than the speeds measured at 1 AU. It is suggested that most of the time the low-latitude solar wind completes accelerating within 20 solar radii, but occasionally additional acceleration is present beyond 20 solar radii.