gistfile1.scala

case object One 

trait Zero[A] {
  def zero: A
}

object Zero {
  implicit val oneZero: Zero[One.type] = new Zero[One.type] {
    def zero: One.type = One 
  }
}

trait Monoid[A] {
  def mappend(a: A, b: A): A
  def mzero: A
}

object Monoid {
  implicit val oneMonoid: Monoid[One.type] = new Monoid[One.type] {
    def mappend(a: One.type, b: One.type): One.type = One 
    def mzero: One.type = One 
  }
}

object Main {
  def test[A](a: A)(implicit zero: Zero[A],
                    mon: Monoid[A],
                    ev: Zero[A] =:= Zero[One.type]) = mon.mappend(a, a)

  def main(args: Array[String]) {
    println(test(One))                                                                          
  }
}

Looking back. Zero only makes sense in combination with an operation. Scalaz also corrected this.
Zero as a typeclass does not make sense.