We define an integral of a vector-valued function
$f:\Omega\longrightarrow X$ with respect to a bounded
countably additive vector-valued measure
$\nu:\Sigma\longrightarrow Y$ and investigate its
properties. The integral is an element of $X\check{\otimes}Y$,
and when $f$ is $\nu$-measurable we show that $f$ is
integrable if and only if $\|f\|\in L_{1}(\nu)$. In this case,
the indefinite integral of $f$ is of bounded variation if and
only if $\|f\|\in L_{1}(|\nu|)$. We also define the integral
of a weakly $\nu$-measurable function and show that such a
function $f$ satisfies $x^{*}f\in L_{1}(\nu)$ for all
$x^{*}\in X^{*}$ and is $|y^{*}\nu|$-Pettis integrable for all
$y^{*}\in Y^{*}$.