  We describe Doppler spectroscopy of Bose-Einstein condensates of ytterbium
atoms using a narrow optical transition. We address the optical clock
transition around 578 nm between the ${^1}S_0$ and ${^3}P_0$ states with a
laser system locked on a high-finesse cavity. We show how the absolute
frequency of the cavity modes can be determined within a few tens of kHz using
high-resolution spectroscopy on molecular iodine. We show that optical spectra
reflect the velocity distribution of expanding condensates in free fall or
after releasing them inside an optical waveguide. We demonstrate sub-kHz
spectral linewidths, with long-term drifts of the resonance frequency well
below 1 kHz/hour. These results open the way to high-resolution spectroscopy of
many-body systems.
