As part of an on-going program utilizing magnetically trapped ultracold neutrons (UCNs), we are developing a technique that offers the possibility of improving the precision of the neutron lifetime by more than an order of magnitude. The experiment works by loading an Ioffe-type superconducting magnetic trap with UCNs through inelastic scattering of 0.89 nm neutrons with phonons in superfluid 4He. Trapped neutrons are detected when they decay; charged decay electrons ionize helium atoms in the superfluid resulting in scintillation light that is recorded in real time using photomultiplier tubes. At present, we are installing a larger and deeper superconducting magnetic trap into our apparatus, implementing techniques to reduce background events, and working to increase the neutron decay detection efficiency. We report the status of the construction of the improved apparatus.