Within only four years, the power conversion-efficiencies of solar cells using metal-organic halide materials have skyrocketed in an unprecedented manner, reaching up to 17.9% (as of May, 2014). This unprecedented rise in efficiencies spurs the notion that this technology will in the foreseeable future make the transition into a commercial setting. However, one significant weakness of the perovskite absorbers is their vulnerability to heat and moisture. The hygroscopic nature of some of the material constituents causes a rapid decomposition of the material and thus strongly impacts the stability of photovoltaic devices. In the most common architecture the top layer of a device acts as hole-transporting layer, which selectively transfers photogenerated holes from the perovskite to the top electrode. As the top layer, its secondary function is therefore to protect the sensitive absorber from external stressors such as moisture. Thus far, the best performances have always been achieved using the amorphous hole transporter spiro-OMeTAD (2,2,7,7-tetrakis (N,N-dimethoxyphenylamine)-9,9-spirobifluorene). Yet, due to its intrinsically low hole mobility, it requires reactive doping which is commonly done with Li-TFSI. This dopant strongly undermines the overall stability aspect of spiro-OMeTAD because its lithium constituents are highly reactive and likely to interact with the perovskite structure, and, more importantly, its hygroscopic nature leads to an increased moisture insertion through the spiro-OMeTAD layer. In this work, we demonstrate a completely new hole-transporting structure that is composed of two layers. The first layer is a dense, interconnected network of highly conductive, polymer-functionalized single-walled carbon nanotubes, whereas the second layer is a dense, electronically inert and thermally stable polymer matrix. Charge extraction is exclusively mediated by the carbon nanotubes, whereas the polymer matrix fills gaps and holes in the nanotube network thus preventing shunting due to direct contact between electrode and perovskite. More importantly, however, the polymer matrix is key to giving this structure a high degree of stability against both heat and moisture. We show that employing this hole-transporting double-layer structure on perovskite solar cells can achieve efficiencies of more than 15% while simultaneously achieving an unprecedented degree of protection against degradation, even allowing direct contact of an operational device with water. Allowing for both high efficiency as well as stability, this hole-transporting structure may prove to be a crucial development for perovskite solar cells for becoming a viable alternative to conventional photovoltaics.