We present the results of a study of the accretion rate of planetesimals by a growing proto-Jupiter in the core-accretion model. The purpose of our study is to test the assumptions of Pollack et al. regarding the flux of planetesimals and their encounter velocities with the protoplanet. Using a newly developed code, we have accurately calculated planetesimals trajectories during their passage in the envelope by combining detailed three-body integrations with gas drag. To be consistent with Pollack et al., our calculations do not include the effect of nebular gas. The results point to several new findings. For instance, we find that only 4-5 M ⊕ is accreted in the first 1.5 Myr before the onset of rapid gas accretion and ∼10 M ⊕ is accreted simultaneously during this phase. We also find that mass accretion remains small (0.3-0.4 M ⊕) for ∼1 Myr after this time. This late accretion, together with a rapid infall of gas, could lead to a mixing of accreted material throughout the outer regions, which may explain the enhancement of high-Z material in Jupiter's envelope. Results demonstrate that planetesimal encounters with the protoplanetary envelope become so fast that in most cases, ram pressure breaks them up. As a result, the accretion rate is largely independent of the planetesimals' size and composition. We also carried out some calculations considering nebular gas drag. As expected, the accreted mass of planetesimals depended strongly on their size and composition. In general, nebular gas lowered the amount of accreted planetesimals, although the majority of planetesimals were still accreted during the rapid gas accretion phase.