Recently INTEGRAL/SPI has observed a strong and extended emission resulting from electron-positron annihilation in the Galactic center region, consistent with the Galactic bulge geometry. The positron production rate, estimated to more than 10$^{43}$ per second, is very high and raises a challenging question about the nature of the Galactic positron source. Commonly considered astrophysical positron injectors, namely type Ia supernovae are rare events and fall short to explain the observed positron production rate. In this paper, we study the possibility of Galactic positron production by hypernovae events, exemplified by the recently observed SN2003dh/GRB030329, an asymmetric explosion of a Wolf-Rayet star associated with a gamma-ray burst...