J1640+2224 is a fully-recycled millisecond pulsar (MSP) in a wide, nearly circular binary with a white dwarf (WD) companion. Previous optical and radio observations of this system have reported widely-varying values for the companion mass, with some indicating it is a low-mass helium-core (He-core) WD, and others indicating it is a high-mass carbon-oxygen (CO) WD. Much of the confusion comes from the fact that the orbital period is very close to half a year, which makes it difficult to measure the parallax and orbital parameters from pulsar timing observations. I will present results from a recently submitted paper in which we combine radio interferometry and optical observations of this system in order to measure the WD's mass. We report the first precise parallax measurement for this system based on observations made with the Very Long Baseline Array (VLBA). We use this distance to reanalyze archival HST observations of the WD and determine that the WD mass is >0.4 Sun with >90% confidence, indicating the companion is most likely a CO WD. This presents a challenge to binary evolution models, since the small spin period and wide, nearly circular orbit both point to the system forming in a low-mass X-ray binary (LMXB), in which case the companion should be a He-core WD. If the companion is indeed a CO WD, present binary evolution models cannot explain how this system formed.