The addition of $m$ singlet right-handed neutrinos to the Standard Model leads to radiatively generated mass corrections for the $SU(2)_L $ doublet neutrinos. For those neutrinos which are massless at the tree level after this addition, this implies a small mass generated at the two-loop level via $W^{\pm}$ exchange. We calculate these mass corrections exactly by obtaining an analytic form for the general case of $n$ doublets and $m$ singlets. As a phenomenological application, we consider the $m=1$ case and examine the masses and mixings of the doublet neutrinos which arise as a result of the two-loop correction in the light of experimental data from two sources which may shed light on the question of neutrino masses. These are(a) the neutrino detectors reporting a solar neutrino deficit (and its resolution via Mikheyev-Smirnov-Wolfenstein matter oscillations), and (b) the COBE satellite data on the non-zero angular variations of the cosmic microwave background temperature (and its possible implications for hot dark matter). Within the framework of the extension considered here, which leaves the gauge group structure of the Standard Model intact, we show that it is possible for neutrinos to acquire small masses naturally, with values which are compatible with current theoretical bias and experimental data.

24 pages, Two figures, attached as postscript files at the end, Latex, CTP-TAMU-76/93, MRI-PHY-12/93, LTH-326