  Employing Eggleton's stellar evolution code with an optically thick wind
assumption, we have systematically studied the WD + He star channel of Type Ia
supernovae (SNe Ia), in which a carbon-oxygen WD accretes material from a He
main-sequence star or a He subgiant to increase its mass to the Chandrasekhar
mass. We mapped out the parameter spaces for producing SNe Ia. According to a
detailed binary population synthesis approach, we find that the Galactic SN Ia
birthrate from this channel is $\sim$$0.3\times 10^{-3} {\rm yr}^{-1}$, and
that this channel can produce SNe Ia with short delay times ($\sim45-$140 Myr).
We also find that the surviving companion stars in this channel have a high
spatial velocity ($>$400 km/s) after SN explosion, which could be an
alternative origin for hypervelocity stars (HVSs), especially for HVSs such as
US 708.
