Thanks a lot! I think we're almost there. After playing around with it for a bit here's what I got:
border: solid 1px #000000;
overflow: hidden; /*I'd prefer it not to scroll, this appears to do the same effect as 'overflow: auto;'*/
background-color: #e0e0e0; /*Matches '.first'*/
border-right: solid 1px #000000;
border-left: solid 1px #000000;
<div class="cell first">First</div>
<div class="cell second">Second<br/>Second<br/>Second<br/>Second<br/>Second</div>
Some issues with this are:
1) If div.first were to become taller than div.second, the area underneath div.second would be gray. I think this is okay, though, since I don't expect div.first to ever be the taller one.
2) div.row still takes up as much width as it can, in spite of the 'min-width' attribute. =( I can fix this by adding 'position:absolute;' to div.row's style, but this is not ideal either since multiple div.row elements would appear on top of each other.