A new analysis of the precise experimental astrophysical $S$-factors for the direct capture $^3He(alpha,gamma)^7{rm {Be}}$ reaction [B.S. Nara Singh et al., Phys.Rev.Lett. {bf 93} (2004) 262503; D. Bemmerer et al., Phys.Rev.Lett. {bf 97} (2006) 122502; F.Confortola et al., Phys.Rev. {bf C 75} (2007) 065803 and T.A.D.Brown et al., Phys.Rev. {bf C 76} (2007) 055801] populating to the ground and first excited states of $^7{rm Be}$ is carried out based on the modified two - body potential approach in which the direct astrophysical $S$-factor, $S_{34}(E)$, is expressed in terms of the asymptotic normalization constants for $^3{rm {He}}+alphato ^7{rm {Be}}$ and two additional conditions are involved to verify the peripheral character of the reaction under consideration. The Woods--Saxon potential form is used for the bound ($alpha+^3{rm {He}}$)- state and the $^3{rm {He}}alpha$- scattering wave functions. New estimates are obtained for the indirectly measured, values of the asymptotic normalization constants (the nuclear vertex constants) for $^3{rm {He}}+alphato^7{rm {Be}}(g.s.)$ and $^3{rm {He}}+alphato^7{rm {Be}}(0.429 MeV)$ as well as the astrophysical $S$-factors $S_{34}(E)$ at E$le$ 90 keV, including $E$=0. The values of asymptotic normalization constants have been used for getting information about the $alpha$-particle spectroscopic factors for the mirror ($^7Li^7{rm {Be}}$)-pair.
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