  Let k be a field with cohomological dimension less than 3; we call such
fields low-dimensional. Examples include algebraically closed fields, finite
fields and function fields thereof, local fields, and number fields with no
real embeddings. We determine the 1-column of the motivic Adams-Novikov
spectral sequence over k. Combined with rational information we use this to
compute the first stable motivic homotopy group of the sphere spectrum over k.
Our main result affirms Morel's pi_1-conjecture in the case of low-dimensional
fields. We also determine stable motivic pi_1 in integer weights other than -2,
-3, and -4.
