The Vineyard Auction Trap

A family vineyard is for sale. Its true value $V$ (in millions) is uniformly distributed on $[0, 1]$ and is known to the family but not to you. You submit a single sealed offer $B$; the family sells if and only if $B \ge V$. Under your management the vineyard would be worth $1.5V$. What offer $B$ maximizes your expected profit?