The codeposition of laser-ablated tungsten atoms with neat hydrogen at 4 K forms a single major product with a broad 2500 cm-1 and sharp 1860, 1830, 1782, 1008, 551, and 437 cm-1 absorptions, which are assigned to the WH4(H2)4 complex on the basis of isotopic shifts and agreement with isotopic frequencies calculated by density functional theory. This D2d structured complex was computed earlier to form exothermically from W atoms and hydrogen molecules. Annealing the matrix allows hydrogen to evaporate and the complex to aggregate and ultimately to decompose. Comparison of the H-H stretching mode at 2500 cm-1 and the W-H2 stretching mode at 1782 cm-1 with 2690 and 1570 cm-1 values for the Kubas complex W(CO)3(PR3)2(H2) suggests that the present physically stable WH4(H2)4 complex has more strongly bound dihydrogen ligands. Our CASPT2 calculations suggest a 15 kcal/mol average binding energy per dihydrogen molecule in the WH4(H2)4 complex.