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We have carried out a high-resolution imaging search for the galaxyassociated with the damped Lyman-alpha (DLA) absorber at z=1.859 toward thez_{em}=2.48 quasar QSO 1244+3443, using the HST and the NICMOS. Images wereobtained in the broad filter F160W and the narrow filter F187N with camera 2 onNICMOS with the goal of detecting the rest-frame optical continuum and theH-alpha line emission from the DLA. After PSF subtraction, two weak featuresare seen at projected separations of 0.16-0.24" from the quasar. Parts of thesefeatures may be associated with the DLA absorber, although we cannot completelyrule out that they could be artifacts of the point spread function (PSF). Ifassociated with the DLA, the objects would be ~1-2 h_{70}^{-1} kpc in size withintegrated flux densities of 2.5 and 3.3 mu Jy in the F160W filter, implyingluminosities at lambda_{central}=5600 A in the DLA rest frame of 4.4-5.9 x10^{9} h_{70}^{-2} L_{solar} at z=1.86, for q0=0.5. However, no significantH-alpha line emission is seen from these objects, suggesting low star formationrates (SFRs). Our 3 sigma upper limit on the SFR in the DLA is 1.3 h_{70}^{-2}M_{solar}/yr for q0 = 0.5 (2.4 h_{70}^{-2} M_{solar} yr^{-1} for q0 = 0.1).This together with our earlier result for LBQS 1210+1731 mark a significantimprovement over previous constraints on the star formation rates of DLAs. Acombination of low SFR and some dust extinction is likely to be responsible forthe lack of H-alpha emission. Alternatively, the objects, may be associatedwith the quasar host galaxy. In any case, our observations suggest that the DLAis not a large bright proto-disk, but a compact object or a low-surfacebrightness galaxy. If the two features are PSF artifacts then the constraintson DLA properties are even more severe.Comment: 28 pages, 9 figures. Accepted for publication in The Astrophysical  Journal. Figures are given at a slightly lower resolution here, to decrease  file sizes. The higher resolution versions can be found in the Ap

A Search for the Damped Lyα Absorber atz = 1.86 toward QSO 1244+3443 with NICMOS