We have completed a direct-imaging search for faint companions (FCs) to 23 stars within 13 pc of the Sun using the Hubble Space Telescope Planetary Camera. The strategy of this search changed considerably from that reported in 1996. To maximize the image contrast between potential FCs and a target star's point-spread function, we adopted the F1042M filter (λc~1.02 μm, Δλ~0.04 μm) as the primary bandpass of our search. Although our sensitivity to FCs varied with the brightness of and separation from our target stars, an ultimate 10 σ detection limit of m1042~18 within 17" of the fainter targets was achieved. As the end of the main sequence occurs at M1042~12, this detection limit makes our search for FCs to nearby stars the most sensitive yet published. Despite this great sensitivity, no previously undetected FCs were found. Our survey would have detected all stellar companions within 17" of our target stars, except for any lowest mass companions lying within 0.5"-1" of the brightest (V<1.5) targets. Applying recent models of brown dwarf spectra and evolution, we find that our search was sensitive to young (0.5 Gyr), low-mass (less than 10 MJ) brown dwarf companions to the fainter targets within 5 pc. A brown dwarf with mass 40 MJ and age 5 Gyr would have been detected at separations greater than 5" from Gl 559A (α Centauri A). Our search was not sensitive to 1 Gyr-old brown dwarfs with masses <~5 MJ, nor was it sensitive to 5 Gyr-old brown dwarfs with masses <~10 MJ. On the positive side, we show the first direct image of the triple system GJ 1245ABC in which all three components are well resolved. Multiband photometry of GJ 1245C suggests for that object a spectral type later than M7. The measured positions of GJ 1245AC are discordant with those expected from published photocentric-orbit elements. The observed positions of four other multiple systems (Gl 65AB, Gl 244AB, Gl 280AB, and Gl 866ABC) also differ from those expected from published orbital elements. The discrepancies noted for Gl 280AB are sufficient to resolve the long-standing inconsistency between Procyon's observed luminosity and that derived from the theoretical mass-luminosity relation. F1042M images of the astrometric binary Gl 105A do not reveal the presence of a fourth component, as has been proposed to reconcile the differences between the observed location of the M7 V companion Gl 105C and the predicted separations of the perturbing body from two independent astrometric studies.