Establishing the resonance spectrum of the nucleon with accuracy would provide important new information about the dynamics and degrees of freedom of its constituents. The spectrum and properties of nucleon resonances are a fundamental test of the emerging predictions from Lattice QCD calculations and will guide re finements to QCD-based phenomenological models. Pion photoproduction is an excellent tool to study the nucleon resonance spectrum, as this channel is expected to couple strongly to most resonances. The new generation of measurements for this reaction, of which the measurement presented in this thesis forms a crucial part, will provide a great improvement in the quality of available experimental data. For the photoproduction process in particular, the use of photon beams and targets with high degrees of polarisation, coupled with large acceptance particle detectors is essential for disentangling the spectrum of excited states. There are many nucleon resonances predicted by recent Lattice QCD calculations and by phenomenological nucleon models which are only observed inconsistently by different analyses of the same experimental data or which are not observed at all. It is of upmost importance to establish if this means that the resonances do not exist in nature, reflecting inappropriate degrees of freedom in the theoretical description of the nucleon or if the current experimental measurements have not been sensitive enough. As such, there is a current world effort at modern tagged photon facilities to measure the \complete set" of photoproduction observables necessary to fully constrain the partial wave analyses used to extract the experimental excitation spectrum from the data. This thesis will present the first detailed measurement to date of positive pion photoproduction in the 730-2300 MeV photon energy (1400-2280 MeV centre-of-mass energy) region with a linearly polarised photon beam and a longitudinally polarised proton target with a close-to-complete angular coverage in detection of the reaction products. This unique set up allows for the extraction of the double-polarisation observable, G. The data were taken as part of the g9 experiment at the Thomas Jefferson National Accelerator Facility in Virginia, using a tagged, polarised photon beam and the Frozen Proton Spin Target, FROST, in conjunction with the CEBAF Large Acceptance Spectrometer, CLAS. The results of the study presented here are compared to the sparse existing data set for the G double-polarisation observable along with the current solutions of the the three main partial wave analyses: MAID, SAID and Bonn-Gatchina. Some agreement is obtained with the expectations of these PWA at lower energies, while disagreement at higher energies is clearly evident. This is the energy region where many of the missing resonances are expected to lie. Once incorporated into the MAID, SAID and Bonn-Gatchina models, these new data will provide an important contribution to constraining the amplitudes and therefore the resonance spectrum and properties of the nucleon. The new data will form a central part of the world effort to accurately establish the nucleon excitation spectrum for the first time by achieving the first complete measurement of experimental observables in meson photoproduction.