  The enforcement of the unimodularity condition in a gravity theory by means
of a Lagrange multiplier leads, in general, to inconsistencies upon
quantization. This is so, in particular, when the classic linear splitting of
the metric between the background and quantum fields is used. To avoid the need
of introducing such a Lagrange multiplier while using the classic linear
splitting, we carry out the quantization of unimodular gravity with extra Weyl
symmetry by using Becchi-Rouet-Stora-Tyutin (BRST) techniques. Here, two gauge
symmetries are to be gauge-fixed: transverse diffeomorphisms and Weyl
transformations. We perform the gauge-fixing of the transverse diffeomorphism
invariance by using BRST transformations that involve antisymmetric ghost
fields. We show that these BRST transformations are compatible with the BRST
transformations needed to gauge-fix the Weyl symmetry, so that they can be
combined in a set of transformations generated by a single BRST operator.
Newton's law of gravitation is derived within the BRST formalism we put forward
as well as the Slavnov-Taylor equation.
