  [Abridged] We present a molecular survey of the starless cores L1498 and
L1517B. These cores have been selected for their relative isolation and
close-to-round shape, and they have been observed in a number of lines of 13
molecular species (4 already presented in the first part of this series): CO,
CS, N2H+, NH3, CH3OH, SO, C3H2, HC3N, C2S, HCN, H2CO, HCO+, and DCO+. Using a
physical model of core structure and a Monte Carlo radiative transfer code, we
determine for each core a self-consistent set abundances that fits
simultaneously the observed radial profile of integrated intensity and the
emergent spectrum towards the core center (for abundant species, optically thin
isopologues are used). From this work, we find that L1498 and L1517B have
similar abundance patterns, with most species suffering a significant drop
toward the core center. This occurs for CO, CS, CH3OH, SO, C3H2, HC3N, C2S,
HCN, H2CO, HCO+, and DCO+, which we fit with profiles having a sharp central
hole. The size of this hole varies with molecule: DCO+, HCN, and HC3N have the
smallest holes while SO, C2S and CO have the largest holes. Only N2H+ and NH3
are present in the gas phase at the core centers. From the different behavior
of molecules, we select SO, C2S, and CH3OH as the most sensitive tracers of
molecular depletion. Comparing our abundance determinations with the
predictions from current chemical models we find order of magnitude
discrepancies. Finally, we show how the ``contribution function'' can be used
to study the formation of line profiles from the different regions of a core.
