  We present the production of dual-species Bose-Einstein condensates of
$^{39}\mathrm{K}$ and $^{87}\mathrm{Rb}$. Preparation of both species in the
$\left| F=1,m_F=-1 \right\rangle$ state enabled us to exploit a total of three
Fesh\-bach resonances which allows for simultaneous Feshbach tuning of the
$^{39}\mathrm{K}$ intraspecies and the $^{39}\mathrm{K}$-$^{87}\mathrm{Rb}$
interspecies scattering length. Thus dual-species Bose-Einstein condensates
were produced by sympathetic cooling of $^{39}\mathrm{K}$ with
$^{87}\mathrm{Rb}$. A dark spontaneous force optical trap was used for
$^{87}\mathrm{Rb}$, to reduce the losses in $^{39}\mathrm{K}$ due to
light-assisted collisions in the optical trapping phase, which can be of
benefit for other dual-species experiments. The tunability of the scattering
length was used to perform precision spectroscopy of the interspecies Feshbach
resonance located at $117.56(2)\,\mathrm{G}$ and to determine the width of the
resonance to $1.21(5)\,\mathrm{G}$ by rethermalization measurements. The
transition region from miscible to immiscible dual-species condensates was
investigated and the interspecies background scattering length was determined
to $28.5\,a_\mathrm{0}$ using an empirical model. This paves the way for
dual-species experiments with $^{39}\mathrm{K}$ and $^{87}\mathrm{Rb}$ BECs
ranging from molecular physics to precision metrology.
