  Ultra-light axions (ULAs) with mass less than 10^-20 eV have interesting
behaviors that may contribute to either dark energy or dark matter at different
epochs of the Universe. Its properties can be explored by cosmological
observations, such as expansion history of the Universe, cosmic large-scale
structure, cosmic microwave background, etc. In this work, we study the ULAs
with a mass around 10^-33 eV, which means the ULA field still rolls slowly at
present with the equation of state w=-1 as dark energy. In order to investigate
the mass and other properties of this kind of ULA field, we adopt the
measurements of Type Ia supernova (SN Ia), baryon acoustic oscillation (BAO),
and Hubble parameter H(z). The Markov Chain Monte Carlo (MCMC) technique is
employed to perform the constraints on the parameters. Finally, by exploring
four cases of the model, we find that the mass of this ULA field is about
3x10^-33 eV if assuming the initial axion field phi_i=M_pl. We also investigate
a general case by assuming phi_i< M_pl and find that the fitting results of
phi_i/M_pl are consistent with or close to 1 for the datasets we use.
