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This article has been accepted for publication in Monthly Notices of the Royal Astronomical Soicety © 2018 The Author(s). Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved ; We present results of the first dynamical stream fits to the recently discovered Tucana III stream. These fits assume a fixed Milky Way potential and give proper motion predictions, which can be tested with the upcoming Gaia Data Release 2 (DR2). These fits reveal that Tucana III is on an eccentric orbit around the Milky Way and, more interestingly, that Tucana III passed within 15 kpc of the Large Magellanic Cloud (LMC) approximately 75 Myr ago. Given this close passage, we fit the Tucana III stream in the combined presence of the Milky Way and the LMC. We find that the predicted proper motions depend on the assumed mass of the LMC and that the LMC can induce a substantial proper motion perpendicular to the stream track. A detection of this misalignment will directly probe the extent of the LMC's influence on our Galaxy, and has implications for nearly all methods which attempt to constraint the Milky Way potential. Such a measurement will be possible with the upcoming Gaia DR2, allowing for a measurement of the LMC's mass. ; DE and VB acknowledge that the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007- 2013) / ERC Grant Agreement no. 308024. EB acknowledges financial support from the European Research Council (StG-335936). ABP acknowledges generous support from the George P. and Cynthia Woods Institute for Fundamental Physics and Astronomy at Texas A&M University. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro- Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacâo Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnología e Inovacâo, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey