  Utilizing the James Webb Space Telescope Early Release NIRCam Observations,
we perform a weak-lensing analysis of the massive galaxy cluster SMACS
J0723.3-7327 ($z=0.39$). We investigate the spatial variation of the PSF from
the stars in the mosaic image. Our measurements show that the PSF for both
modules has very small spatial and temporal variation with average complex
ellipticity components of $e_1=0.007\pm0.001$ and $e_2=0.029\pm0.001$ in the
observed north-up reference frame. We create PSF models through a principal
component analysis of the stars and show that they properly account for the
ellipticity of the PSF with residual shapes of $e_1=(0.3\pm3.5)\times10^{-4}$
and $e_2=(1.8\pm4.0)\times10^{-4}$. We select background galaxies by their
photometric redshift and measure galaxy shapes by model fitting. Our
weak-lensing source catalog achieves 215 galaxies arcmin$^{-2}$. We map the
projected mass density of SMACSJ0723 and detect the cluster with a peak
significance of $12.2\sigma$. The mass distribution is found to elongate in the
east-west direction with an extension to the northeast edge of the field of
view where a candidate substructure is found in the Chandra X-ray imaging. We
fit the tangential shear with a Navarro-Frenk-White model and estimate the mass
of the cluster to be $M_{500}=7.9\pm1.1\times10^{14}$ M$_{\odot}$
($M_{200}=11.4\pm1.5\times10^{14}$ M$_\odot$ ), which agrees with existing mass
estimates. Combining the multiwavelength evidence from literature with our
weak-lensing analysis, we hypothesize that SMACSJ0723 is observed near first
pericenter passage and we identify candidate radio relics.
