I want to make half border of a circle which fades out at the end, like this:

I managed to create a border that fades out to the bottom like this:
#cont{
background: -webkit-linear-gradient(green, #fff);
width: 300px;
height: 300px;
border-radius: 1000px;
padding: 10px;
}
#box{
background: black;
width: 300px;
height: 300px;
border-radius: 1000px;
}
<div id="cont">
<div id="box"></div>
</div>
However, this circle does not fade out at 50% but on the bottom. Also the border does not become thinner. How can I achieve this?