We examine the inner mass distribution of the relaxed galaxy cluster Abell 383 in deep 16-band HST/ACS+WFC3 imaging taken as part of the CLASH multi-cycle treasury program. Our program is designed to study the dark matter distribution in 25 massive clusters, and balances depth with a wide wavelength coverage to better identify lensed systems and generate precise photometric redshifts. This information together with the predictive strength of our strong-lensing analysis method identifies 13 new multiply-lensed images and candidates, so that a total of 27 multiple-images of 9 systems are used to tightly constrain the inner mass profile, $dlog Sigma/dlog rsimeq -0.6pm 0.1$ (r<160 kpc). We find consistency with the standard distance-redshift relation for the full range spanned by the lensed images, 1.01<z<6.03, with the higher redshift sources deflected through larger angles as expected. The inner mass profile derived here is consistent with the results of our independent weak-lensing analysis of wide-field Subaru images, with good agreement in the region of overlap. The overall mass profile is well fitted by an NFW profile with M_{vir}=(5.37^{+0.70}_{-0.63}pm 0.26) x 10^{14}M_{odot}/h and a relatively high concentration, c_{vir}=8.77^{+0.44}_{-0.42}pm 0.23, which lies above the standard c-M relation similar to other well-studied clusters. The critical radius of Abell 383 is modest by the standards of other lensing clusters, r_{E}simeq16pm2arcsec (for z_s=2.55), so the relatively large number of lensed images uncovered here with precise photometric redshifts validates our imaging strategy for the CLASH survey. In total we aim to provide similarly high-quality lensing data for 25 clusters, 20 of which are X-ray selected relaxed clusters, enabling a precise determination of the representative mass profile free from lensing bias. (ABRIDGED)
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