  Levinson's theorem for the Dirac equation is known in the form of a sum of
positive and negative energy phase shifts at zero momentum related to the total
number of bound states. In this letter we prove a stronger version of
Levinson's theorem valid for positive and negative energy phase shifts
separately. The surprising result is, that in general the phase shifts for each
sign of the energy do not give the number of bound states with the same sign of
the energy (in units of $\pi$), but instead, are related to the number of bound
states of a certain Schr\"odinger equation, which coincides with the Dirac
equation at zero momentum.
