  Constraints on dark energy from current observational data are sensitive to
how distances are measured from Type Ia supernova (SN Ia) data. We find that
flux-averaging of SNe Ia can be used to test the presence of unknown systematic
uncertainties, and yield more robust distance measurements from SNe Ia. We have
applied this approach to the nearby + SDSS +ESSENCE +SNLS +HST set of 288 SNe
Ia, and the ``Constitution''of set 397 SNe Ia. Combining the SN Ia data with
CMB data from WMAP five year observations, the Sloan Digital Sky Survey baryon
acoustic oscillation measurements, the data of 69 gammay-ray bursts, and the
Hubble constant measurement from the HST project SHOES, we measure the dark
energy density function X(z)=\rho_X(z)/\rho_X(0) as a free function of redshift
(assumed to be a constant at z>1 or z>1.5). Without flux-averaging of SNe Ia,
the combined data using the ``Constitution'' set of SNe Ia seem to indicate a
deviation from a cosmological constant at ~95% confidence level at 0< z <0.8;
they are consistent with a cosmological constant at ~68% confidence level when
SNe Ia are flux-averaged. The combined data using the nearby +SDSS +ESSENCE
+SNLS +HST data set of SNe Ia are consistent with a cosmological constant at
68% confidence level with or without flux-averaging of SNe Ia, and give dark
energy constraints that are significantly more stringent than that using the
``Constitution'' set of SNe Ia. Assuming a flat universe, dark energy is
detected at >98% confidence level for z<= 0.75 using the combined data with 288
SNe Ia from nearby + SDSS+ ESSENCE +SNLS +HST, independent of the assumptions
about X(z>1). We quantify dark energy constraints without assuming a flat
universe using the dark energy Figure-of-Merit (FoM) for both $X(z)$ and a dark
energy equation-of-state linear in the cosmic scale factor.
