We present new time-resolved photometric observations of the bright trans-Neptunian object (20000) Varuna and use them to study the rotation period, shape, and color. In observations from 2001 February and April, we find a best-fit two-peaked light curve with period 6.3442+/-0.0002 hr. The peak-to-peak photometric range in the R band is 0.42+/-0.02 mag. We find no rotational variation in colors over the 0.45 μm<=λ<=0.85 μm wavelength range. From the short, double-peaked period and large amplitude, we suggest that Varuna is an elongated, body, perhaps close in shape to one of the Jacobi ellipsoids. If so, the ratio of the axes projected into the plane of the sky is 1.5:1 and the density is near 1000 kg m-3. Varuna may be a rotationally distorted rubble pile, with a weak internal constitution due to fracturing by past impacts. The high specific angular momentum implied by our observations and recent detections of binary trans-Neptunian objects both point to an early, intense collisional epoch in which large trans-Neptunian objects were ~100 times more abundant than now. In order to maintain a cosmochemically plausible rock:ice mass ratio of ~0.5, Varuna must be internally porous.