The Belle II experiment is currently under construction in the KEK laboratory in Tsukuba, Japan. It will record events produced by the world's only second-generation B factory, the SuperKEKB collider, an asymmetric electron-positron storage ring with 3 km circumference. Neither the detector of the predecessor experiment Belle nor its reconstruction software would have been able to deal with the amount of events expected and their properties. Therefore the entire event reconstruction software was redesigned from scratch, while crucial parts of the Belle detector were improved and rebuilt to form the new Belle II detector. This thesis describes the VerteX Detector Track Finder (VXDTF). Its task is to reconstruct tracks in the innermost tracking detector, the VXD, a silicon based detector consisting of two parts, the PXD with two layers of pixel sensors using DEPFET technology, and the SVD with four layers of double-sided silicon strip sensors. The potential amount of data recorded by the PXD with its 8 million pixels with 8-bit depth each, read out with about 50,000 readout frames per seconds, cannot be stored, unless a data reduction by a factor of about 10 can be performed. To this end, the VXDTF will be deployed to reconstruct tracks in the SVD in real time. These tracks are then extrapolated to the PXD where they define regions of interest, that are then read out. The focus of the VXDTF therefore lies on full event reconstruction in the SVD only, with an additional emphasis on low-momentum tracking down to a transverse momentum of 50 MeV/c. Such low-momentum tracks can be reconstructed only by a track finder operating in the innermost tracking detector. Given the small number of sensor layers in the SVD, the VXDTF hast to deal with little redundancy, with the stochastic disturbances due to material effects that are particularly strong for low-momentum tracks, and with the increased background level caused by the high luminosity of the collider. This thesis describes the detector Belle II, the machine SuperKEKB and their features; illustrates how typical events recorded by the SVD look like; describes the details of the VXDTF implementation; analyzes its performance on simulated events of Upsilon decays with realistic background; reports its performance at a combined beam test where PXD and SVD sensors were jointly tested in an electron beam at the DESY laboratory. Finally, it describes the next steps regarding the VXDTF and its successor, the VXDTF 2, which is also described in this thesis. The performance of a current version of the VXDTF can be summarized as follows. The reconstruction efficiency of the VXDTF for Upsilon events with full background included is about 87% for all simulated tracks creating enough hits in the SVD to be able to reconstructed. For tracks with a transverse momentum of 100 MeV/c the efficiency is still above 80% and for 50 MeV/c it is about 65%.