  We report the results of a new measurement of spin structure functions of the
deuteron in the region of moderate momentum transfer ($Q^2$ = 0.27 -- 1.3
(GeV/c)$^2$) and final hadronic state mass in the nucleon resonance region ($W$
= 1.08 -- 2.0 GeV). We scattered a 2.5 GeV polarized continuous electron beam
at Jefferson Lab off a dynamically polarized cryogenic solid state target
($^{15}$ND$_3$) and detected the scattered electrons with the CEBAF Large
Acceptance Spectrometer (CLAS). From our data, we extract the longitudinal
double spin asymmetry $A_{||}$ and the spin structure function $g_1^d$. Our
data are generally in reasonable agreement with existing data from SLAC where
they overlap, and they represent a substantial improvement in statistical
precision. We compare our results with expectations for resonance asymmetries
and extrapolated deep inelastic scaling results. Finally, we evaluate the first
moment of the structure function $g_1^d$ and study its approach to both the
deep inelastic limit at large $Q^2$ and to the Gerasimov-Drell-Hearn sum rule
at the real photon limit ($Q^2 \to 0$). We find that the first moment varies
rapidly in the $Q^2$ range of our experiment and crosses zero at $Q^2$ between
0.5 and 0.8 (GeV/c)$^2$, indicating the importance of the $\Delta$ resonance at
these momentum transfers.
