Fission cross sections for the systems $^{169}\mathrm{Tm}$ + $^{11}\mathrm{B}$, $^{175}\mathrm{Lu}$ + $^{11}\mathrm{B}$, and $^{12}\mathrm{C}$, $^{174}\mathrm{Y}$ + $^{12}\mathrm{C}$, $^{182}\mathrm{W}$ + $^{12}\mathrm{C}$, $^{165}\mathrm{Ho}$ + $^{14}\mathrm{N}$, and $^{159}\mathrm{Tb}$ + $^{19}\mathrm{F}$ have been measured for heavy-ion bombarding energies up to 10.4 MeV per nucleon. The experimental technique consisted of counting coincident fission-fragment pairs with two gold-surface-barrier silicon-diode detectors. For the above systems, fission takes place only in reactions in which a compound nucleus is formed between the incident projectile and the target nucleus. Values of the compound-nucleus cross sections for these reactions are estimated from other data in order to account for surface reactions which occur in heavy-ion bombardment. The difference between the cross section for compoundnucleus formation and that for fission is assumed to be equal to the cross section for neutron-evaporation products. The ratio of the fission cross section to that for neutron evaporation is then taken to be equal to $〈\frac{{\ensuremath{\Gamma}}_{f}}{{\ensuremath{\Gamma}}_{n}}〉$, the ratio of the level widths for the two competing processes averaged over the various reaction channels. A theoretical fit to the $〈\frac{{\ensuremath{\Gamma}}_{f}}{{\ensuremath{\Gamma}}_{n}}〉$ values is obtained for the low-energy region of the excitation function, where first-chance fission is highly probable. We find the ratio of the level-density parameter for fission to that for neutron emission to be 1.2 \ifmmode\pm\else\textpm\fi{} 0.1, and values for the fission barrier to be in agreement with those predicted by Myers and Swiatecki.