  Let $\overline{\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted
stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space.
These are compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively,
obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq
1$ to the markings; they are defined over $\mathbb{Z}$, and therefore over any
field. We study the first order infinitesimal deformations of
$\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$. In particular,
we show that $\overline{M}_{0,A[n]}$ is rigid over any field, if $g\geq 1$ then
$\overline{\mathcal{M}}_{g,A[n]}$ is rigid over any field of characteristic
zero, and if $g+n > 4$ then the coarse moduli space $\overline{M}_{g,A[n]}$ is
rigid over an algebraically closed field of characteristic zero. Finally, we
take into account a degeneration of Hassett spaces parametrizing rational
curves obtained by allowing the weights to have sum equal to two. In
particular, we consider such a Hassett $3$-fold which is isomorphic to the
Segre cubic hypersurface in $\mathbb{P}^4$, and we prove that its family of
first order infinitesimal deformations is non-singular of dimension ten, and
the general deformation is smooth.
