A simple chemical precipitation technique is reported for the synthesis of a hybrid nanostructure of single-wall carbon nanotubes (SWCNT) and titania ($TiO_2$) nanocrystals of average size 5 nm, which may be useful as a prominent photocatalytic material with improved functionality. The synthesized hybrid structure has been characterized by transmission electron microscopy (HRTEM), energy-dispersive X-ray analysis (EDAX), powder X-ray diffractometry (XRD), Fourier transform infrared spectroscopy (FTIR) and Raman spectroscopy. It is clearly revealed that nearly monodispersed titania nanocrystals (anatase phase) of average size 5 nm decorate the surfaces of SWCNT bundles. The UV-vis absorption study shows a blue shift of 16 nm in the absorbance peak position of the composite material compared to the unmodified SWCNTs. The photoluminescence study shows a violet-blue emission in the range of 325-500 nm with a peak emission at 400 nm. The low temperature electrical transport property of the synthesized nanomaterial has been studied between 77-300 K. The DC conductivity shows semiconductor-like characteristics with conductivity increasing sharply with temperature in the range of 175-300 K. Such nanocomposites may find wide applications as improved photocatalyst due to transfer of photo-ejected electrons from $TiO_2$ to SWCNT, thus reducing recombination, with the SWCNT scaffold providing a firm and better positioning of the catalytic material.