We report High Energy Transient Explorer 2 (HETE-2) Wide Field X-Ray Monitor/French Gamma Telescope observations of the X-ray flash XRF 020903. This event was extremely soft: the ratio log(SX/Sγ)=0.7, where SX and Sγ are the fluences in the 2-30 and 30-400 keV energy bands, is the most extreme value observed so far by HETE-2. In addition, the spectrum has an observed peak energy of Eobspeak<5.0 keV (99.7% probability upper limit), and no photons were detected above ~10 keV. The burst is shorter at higher energies, which is similar to the behavior of long gamma-ray bursts (GRBs). We consider the possibility that the burst lies at very high redshift and that the low value of Eobspeak is due to the cosmological redshift, and show that this is very unlikely. We find that the properties of XRF 020903 are consistent with the relation between the fluences S(7-30 keV) and S(30-400 keV), found by Barraud et al. for GRBs and X-ray-rich GRBs, and are consistent with the extension by a decade of the hardness-intensity correlation found by the same authors. Assuming that XRF 020903 lies at a redshift z=0.25, as implied by the host galaxy of the candidate optical and radio afterglows of this burst, we find that the properties of XRF 020903 are consistent with an extension by a factor ~300 of the relation between the isotropic-equivalent energy Eiso and the peak Epeak of the νFν spectrum (in the source frame of the burst) found by Amati et al. for GRBs. The results presented in this paper therefore provide evidence that X-ray flashes (XRFs), X-ray-rich GRBs, and GRBs form a continuum and are a single phenomenon. The results also impose strong constraints on models of XRFs and X-ray-rich GRBs.