In recent observations with the Rossi X-Ray Timing Explorer, we have detect
ed two simultaneous quasi-periodic oscillation (QPO) peaks in the low-mass
X-ray binary and atoll source 4U 1735-44. The lower and higher frequency QP
Os have frequencies varying between 632 and 729 Hz and 982 and 1026 Hz, res
pectively. The fractional rms amplitudes are 3.7%-8.1% and 5.0%-5.8%. The f
requency separation between the two QPOs changes from 341 +/- 7 to 296 +/-
12 Hz. The inferred mass accretion rate during our observations is relative
ly low compared to that during the previous observations, in which only a s
ingle QPO was present. There is weak evidence that the frequency of the QPO
s correlates with the mass accretion rate, as observed in other binaries. F
ive X-ray bursts were recorded with no detectable oscillations with upper l
imits for the rms fraction of 4%-13%.