A torsion-bar antenna (TOBA) is a low-frequency terrestrial gravitational wave (GW) antenna which consists of two orthogonal bar-shaped test masses. We upgraded the prototype TOBA and achieved the strain sensitivity $10^{-10}$ Hz$^{-1/2}$ at around 1 Hz. We operated the upgraded TOBA (called the "Phase-II TOBA") located at Tokyo in Japan for 22.5 hours and performed an all-sky coherent search for continuous GWs using the $\mathcal{F}$-statistic. We place upper limits on continuous GWs from electromagnetically unknown sources in the frequency range from 6 Hz to 7 Hz with the first derivative of frequency less than $7.62 \times 10^{-11}$ Hz/s using data from the TOBA. As a result, no significant GW signals are found in the frequency band 6-7 Hz. The most stringent upper limit upper limit on the dimensionless GW strain with 95% confidence level in this band is $3.6 \times 10^{-12}$ at 6.84 Hz.
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