Single-Wall Carbon Nanotubes (SWNTs) are among the very few candidates for single-photon sources operating in the telecom bands since they exhibit large photon antibunching up to room temperature [1][2][3][4][5]. However, coupling a nanotube to a photonic structure is highly challenging because of the random location and emission wavelength in the growth process [6][7][8][9]. Here, we demonstrate the realization of a widely tunable single-photon source by using a carbon nanotube inserted in an original repositionable fiber micro-cavity: we fully characterize the emitter in the free-space and subsequently form the cavity around the nanotube. This brings an invaluable insight into the emergence of quantum electrodynamical effects. We observe an efficient funneling of the emission into the cavity mode with a strong sub-Poissonian statistics together with an up to 6-fold Purcell enhancement factor. By exploiting the cavity feeding effect on the phonon wings, we locked the single-photon emission at the cavity frequency over a 4 THz-wide band while keeping the mode width below 80 GHz. This paves the way to multiplexing and multiple qubit coupling.Coupling a carbon nanotube to a photonic resonator in a reliable way is highly desirable both for technological developments in view of quantum cryptography or quantum computation and for academical studies since nanotubes behave like an original nano-emitter showing an hybrid 1D-0D electronic behavior [10]. In addition, the low-cost, the high integrability and the possible electrical excitation of nanotubes [11] are attractive assets in such perspectives. However, due to the lack of control of the current growth or deposition processes, current attempts rely on random spectral and spatial matching between a resonator (micro-discs [12] or photonic crystals [13]) and randomly deposited nanotubes, leading to a very limited fabrication yield. This constrain becomes especially stringent when a high coupling between the emitter and the cavity is sought, which requires narrow spectral features.In this work, we propose an original approach where the nanotube is fully characterized in free-space by regular low-temperature micro-photoluminescence (micro-PL) spectroscopy and where a micro-cavity is subsequently formed around the emitter by approaching a concave dielectric mirror micro-engineered at the apex of an optical fiber. This geometry brings an unprecedented flexibility giving a built-in spectral and spatial matching, together with excellent quality factors and mode volumes [14]. Individual carbon nanotubes embedded in a polystyrene matrix were coupled to the cavity resulting in a strong brightening of the nanotube of more than an order of magnitude, bringing evidence for the relevance of exploiting cavity quantum electrodynamical (CQED) effects to enhance the photonic properties of carbon nanotubes. By means of time-resolved measurements we were able to investigate directly the cavityenhanced emission rate and found a corresponding Purcell factor F p of up to 5. In the same tim...