Evolutionary scenarios suggest that the progenitor of the new binary pulsar J0737-3039B \cite{ref1,ref2} was a He-star with $M > 2.1-2.3~\Ms$ \cite{ref3,ref4}. We show that this case implies that the binary must have a large ($>120$ km/s) center of mass velocity. However, the location, $\sim 50$ pc from the Galactic plane, suggests that the system has, at high likelihood, a significantly smaller center of mass velocity and a progenitor more massive than 2.1~$\Ms$ is ruled out (at 97% c.l.). A progenitor mass around 1.45~$\Ms$, involving a new previously unseen gravitational collapse, is kinematically favored. The low mass progenitor is consistent with the recent scintillations based velocity measurement of 66$\pm 15$ km/s \cite{ref12new} (and which also rules out the high mass solution at 99% c.l.) and inconsistent with the higher earlier estimates of 141$\pm 8.5$ km/s \cite{ref11new}. Direct proper motion measurements, that should be available within a year or so, should better help to distinguish between the two scenarios.

4 pages, 2 figures, submitted to PRL