We have observed the HCO+ J = 3-2 line toward 17 starless cores selected fr
om the list of Ward-Thompson and coworkers. Six of these cores have line as
ymmetries indicative of collapse. The excess of blue-skewed profiles over r
ed-skewed profiles is at least as large as that found in samples of Class 0
and early Class I sources. The observed line profiles have the same narrow
line widths and small peak temperatures predicted for young sources in evo
lutionary models, but the blue/red ratios, like those of older sources, are
higher than models predict. The infall signature also occurs over large sc
ales, suggesting that these cores have overall inward motions. We have divi
ded these starless cores into two groups based on the continuum photometry
of Ward-Thompson and coworkers and our HCO+ data. We find stronger HCO+ emi
ssion among the cores detected in the submillimeter, and all the blue-skewe
d line profiles are in this group, supporting the suggestion of Ward-Thomps
on and coworkers that these are the preprotostellar cores.