With new radioactive isotope accelerators coming online in the next decade, the problem of extracting reliable nuclear structure information from reactions with unstable nuclei deserves considerable attention. A method has been proposed to determine spectroscopic factors (SFs) using the asymptotic normalization coefficient (ANC) to fix the external contribution of a nonperipheral reaction, reducing the uncertainty in the SF. The ¹⁵C[left right arrow]¹⁴C+n system was chosen as a test case for this new method. The direct neutron capture rate on ¹⁴C is important for a variety of topics of interest in astrophysics, and the ANC for ¹⁵C[left right arrow]¹⁴C+n was also used to calculate this reaction rate. The objective of the first part of this work was to find the ANC for ¹⁵C[left right arrow]¹⁴C+n. This was done in two independent experiments. First, the heavy ion neutron transfer reaction ¹³C(¹⁴C,¹⁵C)¹²C was measured at 12 MeV/nucleon. Second, the inverse kinematics reaction d(¹⁴C,p)¹⁵C was measured using the new Texas Edinburgh Catania Silicon Array (TECSA). The next phase of the experimental program was to measure a reaction with a non-negligible interior contribution, for which ¹⁴C(d,p)¹⁵C at 60 MeV deuteron energy was used. This reaction turned out to be more peripheral than anticipated, and as a result, the ANC for the ground state was extracted from this measurement as well. The final results for the three measurements are C²2s1/2 = 1.96±0.16 fm⁻¹ for the ground state and C²1d5/2 = (4.23±0.38)·10⁻³ fm⁻¹ for the first excited state. Because the 60 MeV ¹⁴C(d,p)¹⁵C reaction turned out to have a very weak dependence on the interior, the SF could not be determined for the ¹⁴C+n ground state in ¹⁵C using the new method. A lower limit of 1.05 was found for the first excited state. It is possible that other reactions might turn out to be more suitable for this method, however, the difficulty encountered at this relatively high deuteron energy highlights a substantial problem likely to be seen in other applications. Using the ANCs determined in this work, the astrophysical ¹⁴C(n,γ)¹⁵C reaction rate was calculated. The resulting value for the cross section for capture to the ground state at 23 keV was σgs(23 keV)=5.1±0.4 μb and to the first excited state was σexc(23 keV)=0.2±0.02 μb.