  Using an ultrastable continuous-wave laser at 580 nm we performed spectral
hole burning of Eu$^{3+}$:Y$_{2}$SiO$_{5}$ at very high spectral resolution.
Essential parameters determining the usefulness as a "macroscopic" frequency
reference: linewidth, temperature sensitivity, long-term stability were
characterized, using a H-maser stabilized frequency comb. Spectral holes with
linewidth as low as 6 kHz were observed and the upper limit of the drift of the
hole frequency was determined to be on the order of 5$\pm$3 mHz/s. We discuss
necessary requirements for achieving ultra-high-stability in laser frequency
stabilization to these spectral holes.
