The age and formation mechanism of the Moon's lithospheric magnetic anomalies can give insight into the age and strength of the Moon's past global magnetic field. Here we focus on the magnetic anomaly Reiner Gamma, which is associated with a bright sinuous pattern on the surface known as a lunar swirl. We propose that portions of Reiner Gamma's magnetic source bodies were demagnetized by the heat from volcanic domes in the Marius Hills. We model one of the domes as an intrusive magma body and calculate its subsurface thermal evolution to constrain the volume and burial depth of the original magnetized source bodies. We infer that their burial depths do not exceed ~2 km, which is lower than the basalt thicknesses in this region (~3–4 km) and suggests that the source bodies were emplaced during the Moon's major episode of mare volcanism. The source depths and our inferred magnetization magnitudes are consistent with the magnetic source bodies being magnetized by a strong dynamo on the Moon for at least thousands of years at some point between ~3.5–3.9 Ga. In turn, the implied lack of a strong field during demagnetization implies the Moon's dynamo may have been merely episodically strong, or had permanently weakened at the time of dome formation between ~1.0–3.3 Ga.