  This series of papers investigates the dynamic interior of a quiescent
prominence revealed by recent {\it Hinode} and {\it SDO/AIA} high-resolution
observations. This first paper is a study of the static equilibrium of the
Kippenhahn-Schl\"{u}ter diffuse plasma slab, suspended vertically in a bowed
magnetic field, under the frozen-in condition and subject to a theoretical
thermal balance among an optically-thin radiation, heating, and field-aligned
thermal conduction. The everywhere-analytical solutions to this nonlinear
problem are an extremely restricted subset of the physically admissible states
of the system. For most values of the total mass frozen into a given bowed
field, force-balance and steady energy-transport cannot both be met without a
finite fraction of the total mass having collapsed into a cold sheet of zero
thickness, within which the frozen-in condition must break down. An exact,
resistive hydromagnetic extension of the Kippenhahn-Schl\"{u}ter slab is also
presented, resolving the mass-sheet singularity into a finite-thickness layer
of steadily-falling dense fluid. Our hydromagnetic result suggests that the
narrow, vertical prominence $H_{\alpha}$ threads may be falling across magnetic
fields, with optically-thick cores much denser and ionized to much lower
degrees than conventionally considered. This implication is discussed in
relation to (i) the recent {\it SDO/AIA} observations of quiescent prominences
that are massive and yet draining mass everywhere in their interiors, (ii) the
canonical range of $5-60 G$ determined from spectral-polarimetric observations
of prominence magnetic fields over the years and (iii) the need for a more
realistic multi-fluid treatment.
