This paper presents an absorption-line study of θ1 C Orionis (HD 37022), based on IUE data. The line of sight, as determined by O'Dell et al. (1993), is thought to consist of two major cloud complexes: the ``neutral lid'' and some foreground material. A key challenge was to improve the S/N for IUE spectra, and this is discussed in some detail. Column densities for several species are derived in two ways. First, we fit equivalent widths of the interstellar lines on two semiempirical curves of growth (COGs). These COGs are derived from high-resolution ground-based data that resolve the cloud component structure on the line of sight (LOS), and it is assumed that the atomic species follow this structure. Second, we fit each species independently to its own COG, yielding approximate b-values and column densities. Both methods yielded similar results. The Lyα line yields a column density for hydrogen of log N(H I) = 21.60 +/- 0.1 cm-2. Abundances and depletions are calculated for the clouds and are compared to emission-line studies of the Orion nebula itself. The discrepancies between the two regions imply grain core destruction in the H II region. Also, the depletions are compared to other sight lines with different far-ultraviolet extinction behavior. It is found that depletions are not strongly dependent on far-UV extinction except for Mg and possibly Si and P. These elements seem slightly less depleted for sight lines with flat far-UV extinction. Ionization balance gives an electron density of ne ~ 5 × 10-3 cm-3 for the foreground clouds, which in turn yields a lower limit to the hydrogen density of nH > 9 cm-3.