  We present the first sub-mJy ($\approx0.7$ mJy beam$^{-1}$) survey to be
completed below 100 MHz, which is over an order of magnitude deeper than
previously achieved for widefield imaging of any field at these low
frequencies. The high resolution ($15 \times 15$ arcsec) image of the Bo\"otes
field at 34-75 MHz is made from 56 hours of observation with the LOw Frequency
ARray (LOFAR) Low Band Antenna (LBA) system. The observations and data
reduction, including direction-dependent calibration, are described here. We
present a radio source catalogue containing 1,948 sources detected over an area
of $23.6$ deg$^2$, with a peak flux density threshold of $5\sigma$. Using
existing datasets, we characterise the astrometric and flux density
uncertainties, finding a positional uncertainty of $\sim1.2$ arcsec and a flux
density scale uncertainty of about 5 per cent. Using the available deep 144-MHz
data, we identified 144-MHz counterparts to all the 54-MHz sources, and
produced a matched catalogue within the deep optical coverage area containing
829 sources. We calculate the Euclidean-normalised differential source counts
and investigate the low-frequency radio source spectral indices between 54 and
144 MHz, both of which show a general flattening in the radio spectral indices
for lower flux density sources, from $\sim-0.75$ at 144-MHz flux densities
between 100-1000 mJy to $\sim-0.5$ at 144-MHz flux densities between 5-10 mJy,
due to a growing population of star forming galaxies and compact core-dominated
AGN.
