We report millimeter observations of the line of sight to the recently discovered soft gamma repeater SGR 1627-41, which has been tentatively associated with the supernova remnant (SNR) G337.0-0.1. Among the eight molecular clouds along the line of sight to SGR 1627-41, we show that SNR G337.0-0.1 is probably interacting with one of the most massive giant molecular clouds (GMCs) in the Galaxy, at a distance of 11 kpc from the Sun. Based on the high extinction to the persistent X-ray counterpart of SGR 1627-41, we present evidence for an association of this new soft gamma repeater (SGR) with the SNR G337.0-0.1; they both appear to be located on the near side of the GMC. This is the second SGR located near an extraordinarily massive GMC. We suggest that SGR 1627-41 is a neutron star with a high transverse velocity ( approximately 1000 km s-1) escaping the young ( approximately 5000 yr) SNR G337.0-0.1.