Despite the widespread use of positron technology the efficiency of positron utilization is less than 1 %. The low efficiency is primarily due to positron destruction during moderation. Emitted positrons must be filtered through a moderator material to restrict the re-emitted positrons to a useable and defined energy range. The quantum mechanical nature of solidified hydrogen holds the potential to increase the moderator efficiency an order of magnitude over solid neon-the current leader in moderator efficiency. However the characteristics of solid hydrogen as a positron moderator have never been investigated. This paper discusses the design of an experimental cryostat to create solid hydrogen targets for positron beam studies. The design is constrained by the sublimation pressure of the solid hydrogen target exceeding the vacuum pressure of the positron accelerator, and the ability to control hydrogen crystal growth and quality. Thermal control calculations of the test section and heat exchangers are presented.