The first telegraphic connection was opened in September 1859.

INTER-COLONIAL SUBMARINE AND MAGNETIC TELEGRAPH.
The following are the messages which have been interchanged by the Governors of the colonies on the opening of the line :—
Sir Henry Young to Sir Henry Barkly.
Government House. September, 1859.
Sir Henry Young uses the telegraph across Bass’s Straits, for the first time, in order to convoy thanks to Sir Henry Barkly and his government for their liberal aid to this important work, and cordially hopes that it may, lead to many other reciprocal benefits.
———
Sir Henry Barkly to Sir Henry Young. Melbourne, September 29. Sir Henry Barkly hastens, on coming into Melbourne this morning, to acknowledge the receipt of the first telegraphic message from Hobart Town, and congratulates Sir Henry Young on the more intimate and speedy communication established between Tasmania and the other Australian colonies by the submarine cable now successfully laid across Bass’s Straits. In carrying out this important undertaking, it has given the Government and people of Victoria the greatest pleasure to co-operate with their Tasmanian neighbors, with whom they over desire to live in harmony and the interchange of reciprocal advantages, Henry Barkly.
Examiner, 1 October 1859

But it was beset by problems and did not last long.

Hobart Town Daily Mercury, 12 December 1859

OUR MARINE CABLE.
Telegraphic communication between this Colony and Victoria, New South Wale«, and South Australia, is still interrupted, although it is not improbable that it may be re-established during the day. The shore end at Victoria Cove, King’s Island, which has sustained so much damage through its having been laid amongst rocks and kelp of the existence of which no record was made by those employed to survey the bed of the ocean between Cape Otway und King’s Island, has been under-run A piece of spare cable has also been placed on in Hen of the injured portion ; and it has been carried some two miles further round and landed on a fine sandy bottom. The land line will also have to be extended to the new landing place in order to meet the cable. As soon as it was found that the cable had been injured not a moment was lost in having it removed. The contractors, fortunately, had ordered more than was necessary for the completion of their contract or it is probable that the line would have had to remain silent until additional cable could be procured from England. It is of no use lamenting now over the causes which have led to this interruption in our communication with the other Colonies. The survey was made under the auspices of the Victorian government ; and upon that Colony, therefore the expenses consequent upon the inaccuracy of that survey will fall. The contractors were bound to lay down the cable according to that survey. Had they deviated from it their contract would have been void. It certainly does seem almost incredible that a landing place should have been selected which necessitated the Cable lying amongst rocks that could not fail to destroy it almost immediately, and stranger still that these rocks should have been described by those who made the survey as a fine sandy bottom gradually shelving to the shore. Such, however, was the case, and hence the sad injury to this beautiful Cable and the provoking delay in the establishment of Telegraphic communication.

A few years ago we had telegraphic communication with the sister colonies, a station having been fixed here, and from Circular Head by submarine line to George Town, but in consequence of the imperfect nature of the route selected via King’s Inland, and its rocky neighbourhood, we were deprived by the breakage of the cable of all benefit arising from the receipt of rapid intelligence across the Straits.
Mercury, 1 May 1869

Then in May 1869 the connection was again made and this one continued.

THE SUB-MARINE CABLE.
OPENING OF TELEGRAPHIC COMMUNICATION BETWEEN TASMANIA AND VICTORIA.
Telegraphic communication between Low Heads and Cape Schanck was opened on Sunday last, the Bass’ Straits Cable being laid by the agents of the Telegraph Construction and Maintenance Company. The first message transmitted through the cable was addressed by the Governor of Tasmania to the Governor of Victoria, which we published yesterday with the reply from Toorak. The hon. J. M. Wilson, M.L.C., Mayor of Hobart Town, then despatched an identical message of congratulation to the Mayor, of Melbourne, Sydney, Adelaide and Brisbane, to which replies were received in due course yesterday.

The following are the congratulatory messages exchanged on this occasion between the Mayor of this City and those of the chief cities of the. Continental Colonies.-—
Town Hall, Hobart Town, 1st May, 1869.
To the Right Worshipful the Mayor of Melbourne,
The Mayor of Hobart Town offers his congratulations to the Mayor of Melbourne on the resumption of Telegraphic Communication between the capitals of the Australian Continent and of Tasmania by the laying of the Bass’ Straits Cable, and trusts that this auspicious event may prove the means of promoting a closer and more extended social and commercial intercourse between the inhabitants of the whole group of colonies to the common advantage of all. . . JAMES MILNE WILSON, Mayor.

2nd May, 1869.
To the Right Worshipful the Mayor of Hobart Town
The Mayor of Melbourne on behalf of the citizens heartily reciprocates the congratulations and good wishes of the Mayor of Hobart Town on the successful completion of the laying of the Bass’s Straits Cable, and trusts that his wishes for the closer and more extended commercial and social relationship may be fully and completely realised. THOMAS MOUBRAY, Mayor of Melbourne. 3rd May, 1869.
Tasmanian Times, 4 May 1869

(Image cut down from Map Showing Telegraph Lines. at LOC