The biphotons generated from a hot atomic vapor via the process of spontaneous four-wave mixing (SFWM) have the following merits: stable and tunable frequencies as well as linewidth. Such merits are very useful in the applications of long-distance quantum communication. However, the hot-atom SFWM biphoton sources previously had far lower values of generation rate per linewidth, i.e., spectral brightness, as compared with the sources of biphotons generated by the spontaneous parametric down conversion (SPDC) process. Here, we report a hot-atom SFWM source of 960-kHz biphotons with a generation rate of 3.7$\times$ $10^5$ pairs/s. The high generation rate, together with the narrow linewidth, results in a spectral brightness of 3.8$\times$ $10^5$ pairs/s/MHz, which is 17 times of the previous best result with atomic vapors and also better than all known results with all kinds of media. The all-copropagating scheme together with a large optical depth (OD) of the atomic vapor is the key improvement, enabling the achieved spectral brightness to be about one quarter of the ultimate limit. Furthermore, this biphoton source had a signal-to-background ratio (SBR) of 2.7, which violated the Cauchy-Schwartz inequality for classical light by about two-fold. Although an increasing spectral brightness usually leads to a decreasing SBR, our systematic study indicates that both of the present spectral brightness and SBR can be enhanced by further increasing the OD. This work demonstrates a significant advancement and provides useful knowledge in the quantum technology using photons.