Let K be a finite extension of ℚ and let S = {ν} denote the collection of K normalized absolute values on K. Let V K + denote the additive group of adeles over K and let c : V K + → ℝ ≥0 denote the content map defined as c( { a v } ) = ∏ v∈s v( a v ) for { a v }∈ V K + . A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if { a v }∈ V K + with c( { a v } ) > c , then there exists a non-zero element b ∈ K for which v(b)≤v( a v ), ∀v∈ S . Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances.