The first statement follows from Theoreme XI.3.13 in SGA2 (As has been pointed out in the comments to your question.)

I am sorry, I forgot to check that there might be torsion in the Mordell-Weil group.

... if $k$ is the algebraic closure of a prime field you might have to exclude all the $k$-rational points of the parameter space. If $k$ is a finite field this phenomena actually happens (see the previous comments). In characteristic zero there are several results suggesting that the complement of the countable union of the subvarieties contains a $\mathbb{Q}$-rational point.

@MP: If you believe the Tate conjecture (proven for most K3 surfaces over finite fields) then for a surface over a finite field the geometric Picard number has the same parity as the second betti number. The point is that $\overline{\mathbb{F}_p}$ is too small to apply Deligne's version of Noether-Lefschetz. @Donu: You need to assume that $k$ is transcendental over its prime field. From Deligne's definition of generic it follows that there is a countable union of subvarieties in the parameter space, where the conclusion $\rho=1$ does not hold....