  Low-mass X-ray binaries have long been theorised as potential sources of
continuous gravitational-wave radiation, yet there is no observational evidence
from recent LIGO/Virgo observing runs. Even for the theoretically 'loudest'
source, Sco X-1, the upper limit on gravitational-wave strain has been pushed
ever lower. Such searches require precise measurements of the source properties
for sufficient sensitivity and computational feasibility. Collating over 20
years of high-quality spectroscopic observations of the system, we present a
precise and comprehensive ephemeris for Sco X-1 through radial velocity
measurements, performing a full homogeneous reanalysis of all relevant datasets
and correcting previous analyses. Our Bayesian approach accounts for
observational systematics and maximises not only precision, but also the
fidelity of uncertainty estimates - crucial for informing principled
continuous-wave searches. Our extensive dataset and analysis also enables us to
construct the highest signal-to-noise, highest resolution phase-averaged
spectrum of a low-mass X-ray binary to date. Doppler tomography reveals
intriguing transient structures present in the accretion disk and flow driven
by modulation of the accretion rate, necessitating further characterisation of
the system at high temporal and spectral resolution. Our ephemeris corrects and
supersedes previous ephemerides, and provides a factor three reduction in the
number of templates in the search space, facilitating precision searches for
continuous gravitational-wave emission from Sco X-1 throughout the upcoming
LIGO/Virgo/KAGRA O4 observing run and beyond.
